Legless Locomotion: Concept and Analysis

Ravi Balasubramanian

May 2004

The Robotics InstituteCarnegie Mellon University

Pittsburgh, Pennsylvania 15213.

c© Carnegie Mellon University

Abstract

This thesis presents a novel locomotion technique, called legless locomotion, for convex-bodied leggedrobots. The motivation for studying legless locomotion stems from the mobile robot recovery problem,stated simply as what should a robot do when trapped? Legless locomotion is the technique of locomotionfor a legged robot which has no leg-ground contact, but can use the swinging legs to excite body rotations.

Legless locomotion is a result of the interaction between reaction mass-dynamics and robot-ground con-tact kinematics and presents many open research questions. Legless locomotion has many parameters, suchas the robot mass distribution, robot body shape, leg motions, and contact constraints. We use a simple two-legged robot Rocking and Rolling Robot (RRRobot), RHex, a hexapod robot, and simulations to understandhow the phenomena simultaneously interact. We also develop approximate simplified models that give us aqualitative understanding of the system behavior (see figure below). We hope to develop more exact kine-matic reductions of legless locomotion, that will help us control legless locomotion. Finally, we present aclassification structure for mobile robot error due to external physical influences and propose legless loco-motion as a recovery technique for the particular case when a robot gets high-centered, that is, there is noleg-ground contact.

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1 Introduction

This thesis tries to answer two questions:

• How can we analyze a new locomotion technique, legless locomotion?

• How can mobile robots escape when trapped?

We have discovered a new type of locomotion, legless locomotion, where a legged robot translates in theplane by rocking and rolling on its curved stomach without its swinging legs ever touching the ground. Fig. 1shows a sample of this type of locomotion– a sequence of interleaved roll and yaw body rotations that pro-duces predominantly forward translation for a round-bodied robot (these body rotations are produced by legmotions). We propose legless locomotion as an effective locomotion strategy for specific circumstances thata robot may find itself in; for example, when a robot is stuck with no leg-ground contact. Also, we believelegless locomotion has many interesting aspects that will offer an understanding of dynamics systems.

The motivation for legless locomotion comes from instances where robots get high-centered in roughterrain. We define ‘high-centered’ as follows.

Definition 1.1 A robot is high-centered if there is no leg-ground contact (see Fig. 2).

This problem is a particular slice of the nascent locomotion error recovery field, which tries to answer thequestions:

• How do mobile robots get stuck (locomotion error)?

• How can mobile robots escape when stuck (locomotion error recovery)?

By ‘stuck’, we refer to instances where the robot can no longer use its usual locomotion technique becauseof external circumstances. We will explore the locomotion error domain by providing a locomotion errorclassification structure. We project legless locomotion as a technique for escaping from high-centered sit-uations; the strategy will be to translate while high-centered till the robot reaches a situation where it canuse its legs in a conventional sense. In general, there are at least three strategies that a high-centered leggedrobot can follow to escape by rocking and rolling the body using leg motions (see Fig. 3):

1. If the robot has a curved bottom, then the body rotations can incrementally translate the robot till itfalls off the block (rolling legless locomotion, see Fig. 1).

2. If the robot has a jagged bottom, the body rotations can produce translation by taking advantage of thejagged body; the protrusions would act as ‘feet’ (see Fig. 4). By a series of rolling and yawing motions,the robot shifts weight from one ‘foot’ to another and translates (walking legless locomotion).

3. If the robot has a flat bottom, a net frictional force can be produced over the cycle of leg motionsresulting in translation (sliding legless locomotion [27]).

This thesis will focus on the first strategy that uses pure rolling between curved surfaces. This is becauselegged robots such as RHex (see Fig. 2) have light legs that cannot produce sufficient dynamic forces to liftthe robot weight onto the pivot points (required in the second strategy). Also, there will be impacts each timethe weight is transferred between feet. This will produce slip-related translation which is difficult to modelanalytically. The third strategy requires modelling differences in frictional forces over leg motion cycles, adifficult proposition. In contrast, with a rounded body on a flat surface or vice versa, even small torques cancause the robot’s attitude to oscillate, and these oscillations can generate translation when coupled with thenonholonomic contact constraints between the robot body and ground surface. Also, since all interactions

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Figure 1: Body roll and yaw rotations that produce translation. Motions are represented as rotations aboutaxes attached to the body but aligned with the world coordinate frame. There is a local roll rotation betweenpositions (a) and (b) and positions (c) and (d); there is a local yaw rotation between positions (b) and (c) andpositions (d) and (e).

Figure 2: The RHex experimental platform high-centered on a block (http://rhex.net).

of the body with the environment are smooth, there are no discontinuities to model. Thus, locomoting around-bodied robot on its stomach may be effective and is also easier to model than a robot with a jaggedor flat bottom. We will focus on the first strategy, rolling legless locomotion (henceforth referred to just aslegless locomotion, unless otherwise stated), and provide a comparison with the second and third strategyusing experiments.

Legless locomotion is complex, because of the interactions between the system dynamics (moving masses)and kinematics (sphere on plane). How can we study this new locomotion technique? We built a simple pro-totype, Rocking and Rolling Robot (RRRobot) (see Fig. 5), and simulation environments using Lagrangiandynamics. The key idea we will use in this thesis is to find simpler models that capture the important el-ements of mechanical systems and understand the simpler models. This will then give us insight into theinfluence of system dynamics and kinematics in the full, accurate models; for example, we will look at therole of contact constraints and influence of leg motions on body attitude in simplified models and use themto understand their role in the original legless-locomotion model.

What are the most crucial elements of legless locomotion? Legless locomotion is a result of the interactionbetween leg motions, body-attitude dynamics, varying system inertia, leg motions, and the nonholonomiccontact constraints. In this context, many questions arise: Can we find a simple generic model that captures

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Figure 3: Three types of legless locomotion: rolling, walking, and sliding. Rolling legless locomotion (seeFig. 4) and walking legless locomotion (see Fig. 1) ensue when body rotational oscillations are producedby leg motions. Sliding legless locomotion occurs when a net frictional force is produced over the cycle ofplanar body motions caused by leg motions. This thesis focusses on rolling legless locomotion.

Figure 4: Walking legless locomotion: body oscillations can produce locomotion for a robot with bodyprotrusions.

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Figure 5: The RRRobot experimental platform uses halteres to induce body attitude oscillations leading tobody translations.

the above important characteristics? Can the model help in developing control algorithms for legless loco-motion, say, by RRRobot? This thesis will also look at comparing the motions for the generic model wedevelop and the two-legged RRRobot.

Since RRRobot is underactuated and the locomotion is produced out-of-phase body oscillations, planningfor legless locomotion is difficult. The planning problem for RRRobot may be phrased as follows: Given aplanar path, what trajectories must the legs follow to create the desired body rotations? Mechanical systemslike RRRobot typically have dynamics including velocity-related drift terms and acceleration inputs. Also,the equations of motion are complex and do not offer much intuition for control. In contrast, equations ofmotion for kinematic systems offer more intuition, because they do not have velocity-related drift terms andhave velocity inputs. Thus, it is useful to find out if a mechanical system like RRRobot can be reduced to akinematic system [18]; then, we can work with the kinematic reduction, since it is easier. Does a kinematicreduction for legless locomotion always exist? If not, when does it exist? Certain properties like systeminvariances [8] help simplify analysis of complex mechanical systems. This thesis will explore systematicways of finding kinematic models using system invariances.

Locomotion systems are fascinating since they admit a huge scope. They include dynamics and kine-matics elements- while the kinematics element may be specified by the designer, the dynamics element isspecified by nature; for example, a person riding a simple kinematic system like the bicycle has to deal withdynamic properties like friction. There have been many previous efforts to understand kinematic locomotionsystems [5][22] and dynamic locomotion systems [26][14][25]. Most of these systems use direct actuation,where there are contacts between actuated limbs and the environment (kinematic constraint) that produceforces; for example, in walking and running, there is a direct contact between the legs (actuated appendage)and the ground. This thesis focusses on a locomotion technique that uses dynamically coupled actuation.Dynamically coupled actuation is different from direct actuation techniques in that there is an interplay be-tween the dynamics and kinematics, as it occurs in RRRobot’s locomotion– the swinging reaction massesproduce the forces that actuate motion constrained by the slip-free contact. Understanding this couplingbetween the dynamics and kinematics in locomotion is crucial to exploring novel mechanical designs. Thisthesis will present a principled way of exploring a mechanical system’s capabilities using a technique ofcreating simplified models.

After describing the legless locomotion problem in detail in Section 1.1, we present a simple example inSection 1.2 to illustrate our problem. We discuss related and previous work in Section 2 and the problemstatement in Section 3. In Section 4, we propose the technical approach we will follow to address thequestions raised by the problem statement. We summarize the work done to date in Section 5 and lay out

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Figure 6: A car stuck in mud has intermittent frictional contacts.

the schedule for future work in Section 6. We wrap up the proposal with the list of contributions that thisthesis will make to Robotics in Section 7.

1.1 Problem Description

Legged robots offer good rough-terrain mobility, but conventional legged locomotion fails when the robotis stuck. Our interest in this problem arose from experience with RHex [31], a simple and highly mobilehexapod robot (see Fig. 2). Each leg has a single actuated hip joint, which can rotate all the way around.When RHex is high-centered (see Definition 1.1), the only possible means of locomotion is by wigglingthe legs and using them as reaction mass. Suppose there is a single rolling contact between robot andobstacle, because the robot’s stomach or the obstacle is rounded. Then, even small torques can cause therobot’s attitude to oscillate, and these oscillations can generate translations due to the nonholonomic contactconstraints [24] between the body and obstacle. We call this legless locomotion.

Getting high-centered is only a specific instance of robots getting stuck; it is a case where the robot doesnot have the desired contacts with its environment. For example, a legged robot requires slip-free contactbetween each leg and the surface, but when the robot is high-centered, the legs do not have any groundcontact. There are many other ways in which robots can get stuck; for example, consider the case where therobot’s legs get stuck in the a hole– a case where there are undesired contacts. Also, consider the case of acar stuck in snow or mud (see Fig. 6). How can we control the wheels in presence of intermittent contact tomake the car escape? How do we classify all these locomotion errors? Can we find recovery techniques forthe various locomotion errors? Understanding how robots get stuck can make them more robust.

We will focus on the case of high-centered robots, where legless locomotion can be used as a recoverytechnique. To study legless locomotion, we have constructed the simple prototype, RRRobot, which loco-motes on its spherical stomach (see Fig. 5). Since its legs act only as reaction masses, they are more properlycalled halteres, after the dumbbells sometimes used by athletes to give impetus in leaping. Our RRRobotexperiments and simulations [4] suggest that oscillations in body attitude is a practical way of recoveringfrom the high-centered state. The challenge is to find leg trajectories that create body attitude oscillationswhich, when coupled with the nonholonomic contact constraints, cause RRRobot to locomote in the plane.Fig. 1 shows a sequence of interleaved roll and yaw body rotations that produces predominantly forwardtranslation. Similarly, interleaved pitch and yaw body rotations produce predominantly sideways translation

Producing yaw rotations is a crucial common element of both roll-yaw and pitch-yaw body oscillationsthat produce RRRobot locomotion. While gravity acts as an external force for roll and pitch rotations, noexternal forces exist to create yaw rotations. An important element in RRRobot locomotion is the non-constant system inertia which changes with leg configuration (even frame attached to the body). Thus, the

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Figure 7: The Yaw model: the body, pivoted at its body center, can freely rotate about the yaw axis, and thetwo legs with point masses at the distal ends are actuated.

only way to produce yaw rotations is by moving the legs in an out-of-phase manner and exploiting inertiadifferences (see Section 1.2 for more details).

Since producing body yaw is crucial for RRRobot locomotion, we created a model called the Yaw model,a simplification of RRRobot that captures some essential aspects of legless locomotion: underactuation andvariable inertia. The Yaw model is derived from the original RRRobot model by pivoting the body center ata revolute joint aligned with the yaw axis. With the Yaw model, we can study the relationship between legmotions and the body yaw rotational freedom (see [3] and [4] for other simplified models). The next sectiondiscusses the Yaw model.

1.2 Example: The Yaw Model

The Yaw model body is pivoted at its body center and has two masses, each mm, at its ends (see Fig. 7). Eachmassless leg has an actuated hip joint and a point mass ml at the distal end. The Yaw model configurationis represented by q = (θ2,φ1,φ2)T ∈ Q = S1×S1×S1, where θ denotes the body configuration, φ1 leg 1’sjoint configuration, and φ2 leg 2’s joint configuration. The tangent space TqQ of Q is TS1× TS1× TS1.There is no gravity, there are no joint limits, and torques u1 and u2 can be applied at leg joints 1 and 2. TheRiemannian metric g : T Q×T Q→R associated with the Yaw model on Q [1] describing the system kineticenergy is

g(q) =

g11 g12 g13g21 g22 g23g31 g32 g33

, (1)

whereg11 = 2(mm + ml)b2 + mll

2 + 12 ml l

2(cos2φ1 + cos2φ2)),g12 =−mllbsinφ1,g13 = ml lbsinφ2,g21 =−mllbsinφ1,g22 = ml l

2,g23 = 0,g31 = ml lbsinφ2,g32 = 0,

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Table 1: Incremental motion of the yaw model.

Time interval φ1(t) φ2(t) Change in yaw0–1 0→ π/2 0 ε11–2 π/2 0→ π/2 −ε22–3 π/2→ 0 π/2 −ε23–4 0 π/2→ 0 ε1

Net change in yaw 2(ε1− ε2)

g33 = ml l2,

and the twenty seven Christoffel symbols Γijk are computed as

Γijk =

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gil(∂g jl

∂qk +∂gkl

∂q j −∂g jk

∂ql ), (2)

where gi j are the components of the inverse of gi j. Note that there is a symmetry in the Yaw model: g isindependent of yaw rotations, that is, g does not depend on θ .

The Yaw model equations of motion [1] are

qi + Γijkq jqk = uaY i

a, (3)

where Y1 = g−1

010

, and Y2 = g−1

001

are the control vector fields. The input vector fields indicate

that the Yaw model is underactuated; this makes control difficult. Also, assuming zero system initial velocity,then the body must be stationary when the legs are stationary.

Given that the Yaw model is underactuated, can we reach arbitrary Yaw model configuration using legmotions? An important property of the Yaw model that we can use to produce net body yaw using cyclicinterleaved leg motions is the variable inertia. We will move each leg back and forth between extremes of 0and π/2. Each leg will dwell at the extreme for one second and will take one second to transition betweenangles following a cubic spline. The result is a Lie bracket-inspired smoothed square wave, with the two legsout-of-phase with each other (see Fig. 8). This sequence of leg motions yields a net yaw motion, as shownin Table 1. This result can be confirmed by studying the table and thinking about the angular inertia of thesystem. Suppose the body yaw is ε1 during interval t = 0 to t = 1, and is ε2 during interval t = 1 to t = 2.The net yaw during the first two motion segments is different, because the angular inertia varies dependingon whether the leg is stretched out or tucked in. This difference produces net yaw at the end of the motionsequence. We will see a proof in Section 5 that by exploiting these inertia differences, the Yaw model canreach arbitrary configuration. This same property of producing net yaw motion using inertia differences isseen in RRRobot also.

Finding characteristics like that shown in Table 1 for mechanical systems from equations of motionlike (3) is difficult. Also, planning system trajectories using (3) is difficult, because of the velocity-relatedterms and the torque inputs; that is, there is no systematic analytic procedure to find torque inputs to achievea given goal trajectory. We will see in Section 5.5 how to derive a kinematic reduction for the Yaw modeland plan using it. We only present some results of kinematic reduction using symmetry for the Yaw modelhere. A kinematic representation of the yaw model is as follows:

q = X1u1 + X2u2, (4)

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Figure 8: Incremental motion of the yaw model using Lie bracket-inspired leg motions (see Table 1).

where X1 =

−g12/g11

10

, X2 =

−g13/g11

01

, ua ∈ R are the control inputs, and we have assumed

that we have direct control over the leg joint velocities. Clearly, (4) is more compact than (3); also, wecan use existing techniques for kinematic systems to prove controllability for the Yaw model. For example,using simple computations, we find that the Lie bracket [X1,X2] is non-zero and that the kinematic systemcan reach arbitrary configuration. Thus, finding kinematic reductions for dynamic systems is useful. Wehave only discussed kinematic reduction results for the simple Yaw model; we hope this technique of usingsymmetries to develop kinematic reductions will lead us to a general strategy for understanding and findingmotion primitives for complex systems such as the legless locomoting RRRobot. Yaw rotations are a crucialcomponent of RRRobot’s locomotion, and RRRobot’s Lagrangian is invariant to yaw rotations. Exploitingthis symmetry and developing kinematic equations of motion for at least some of RRRobot’s freedoms willhelp simplify analysis of legless locomotion. This will also help develop RRRobot motion primitives andhelp plan paths for RRRobot.

2 Related Work

This section is organized in three parts:

• Locomotion techniques– a summary of the related locomotion systems and the underlying principles.

• Locomotion error and recovery– a summary of existing work that focusses on robot maneuvers thatcan be used in specific circumstances.

• Dynamics study using affine connections– a background of literature and theoretical techniques thataid understanding mechanical systems.

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2.1 Locomotion Techniques

Locomotion, defined as the act or power of moving completely from place to place (locus + motion), isan important problem in robotics. There are two prominent types of locomotion: legged and wheeled.Legged locomotion offers good rough terrain mobility, while wheeled locomotion offers low-energy mobil-ity in structured terrain. Recently, undulatory or wave-like locomotion has become popular in the researchcommunity; a number of interesting aspects of undulatory locomotion have been explored by using thesnakeboard [17] and roller racer [16] as examples.

The locomotion strategy proposed here for RRRobot, legless locomotion, involves the interplay betweenroll-pitch-yaw attitude dynamics and the kinematic nonholonomic contact constraints. Numerous inves-tigators have studied dynamic systems with constraints using Lagrangian dynamics [12] or the energy-momentum method. Lewis et al. [17] study the constrained mechanics of the constant-inertia snakeboard,a modified version of a skateboard in which wheel directions can be controlled. The snakeboard rider loco-motes by twisting his body back and forth, while simultaneously moving the wheel-directions with a suitablephase relationship. Lewis et al. present numerical simulations of snakeboard locomotion using character-istic wheel motions and discuss a general framework for studying mechanical systems with constraints ina coordinate-free form. Zenkov et al. [39] discuss the energy-momentum method for control of dynamicsystems with nonholonomic constraints such as the rattleback, the roller racer, and the rolling disk. Afteridentifying system symmetries, Zenkov et al. use momentum equations to analyze the system. In studyingRRRobot dynamics, we use the Lagrangian method.

While RRRobot’s spatial position and orientation (called the fiber space [8]) are not actuated, the internalconfiguration of the two legs (called the base space) is actuated. Ostrowski [25] presents a general frame-work for studying systems where the fiber space can be represented as a group. Since only the base spaceis actuated, Ostrowski finds a connection relating the base space velocities to the fiber velocities. WhileOstrowski focusses on systems with constant inertias and simple constraints, such as the snakeboard and theHirose snake, RRRobot has a spherical contact with the plane, and its inertias change with configuration.Thus, it is unclear if this framework can be extended to encompass RRRobot behavior.

RRRobot locomotes by rolling its round body without slip on the planar surface. The curvatures of thetwo surfaces and the type of contact between the two surfaces determine the kinematic constraints and,consequently, the relative motion between the two bodies. Montana [23] derives equations for the motion ofthe contact point between two moving rigid bodies using differential geometry. Camicia et al. [11] providean analysis of the nonholonomic kinematics and dynamics of the Sphericle [7], a hollow ball driven ona planar surface by an unicycle placed inside. Bhattacharya and Agrawal [6] present a spherical rollingrobot that locomotes using two orthogonal rotors placed inside. They derive driftless equations of motionusing the conservation of angular momentum and the contact constraints. The Sphericle, Bhattacharya’srobot, and RRRobot have similar nonholonomic contact constraints (see [24] and [21] for more details onnonholonomic constraints).

If we ignore RRRobot’s ground contacts and constraints, the problem reduces to controlling body attitudeusing halteres. Fernandes et al. [13] discuss near-optimal nonholonomic motion planning for coupled bodiesusing Lagrangian dynamics. Given an arbitrary starting point, Fernandes et al. find plans to land a fallingcat on its feet, subject to the angular momentum conservation nonholonomic constraint. RRRobot’s bodyattitude is not directly actuated, but its inertias change with leg position. By repeatedly wiggling the legswhile exploiting the differences in angular inertia, RRRobot may be able to adjust its orientation. In contrast,when spinning reaction wheels are used to control satellite attitude, the inertias of the satellite system do notchange with rotation of reaction wheels [30].

To understand the interactions between leg motions and body attitude changes, we simplify the RRRobot-on-a-plane model by decoupling the body attitude dynamics from the contact kinematics in approximatemodels (see Section 5.2.2 for more details). This approximate approach of splitting the dynamics from

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Figure 9: The Spring Loaded Inverted Pendulum model is used as a template for RHex’s locomotion.

the kinematics is different from the exact kinematic reduction technique of Bullo et al. [9] for mechanicalsystems with constraints. Kinematic reduction is useful because the kinematic reduced system is easierto control using velocity inputs than the unreduced dynamic system using acceleration inputs (see [9] fordetails on controllability properties for reducible systems).

To understand a mechanical system, one needs to understand the underlying physics, sometimes ignor-ing incidental joints and limbs. Full and Koditschek [28] use models called templates to understand bipedlocomotion. A template is a simple model that eliminates the complexity of joints and serves as a guidefor locomotion control. Altendorfer et al. [2] find evidence for the Spring Loaded Inverted Pendulum tem-plate (see Fig. 9) in RHex’s motion. Schmitt and Holmes [33] propose the Lateral Leg Spring template tostudy insect locomotion in the horizontal plane. We will develop a simple one-legged model for leglesslocomotion.

2.2 Locomotion Error and Recovery

There is not much prior work in the field of locomotion error analysis and recovery techniques. Most of theavailable literature looks at error diagnosis using high-level reasoning techniques on sensor data [36], butnot at how robots get stuck in the physical world. There has been some work in finding robot maneuversthat could act as error recovery techniques, although they are not portrayed as error recovery techniques.Saranli [32] presents back-flips as a technique for ‘uprighting’ RHex (see Fig. 2). RHex can walk upside-down also, but uprighting is useful when RHex uses vision to navigate (to keep images oriented properly).Hale et al. [15] present a single-actuated hopping robot that self-orients itself before propulsion (see Fig. 10).Tunstel [35] discusses a genetic programming approach to finding uprighting maneuvers for a nanorover.The algorithm evaluates the maneuver fitness (quality) using the power consumed, the time elapsed, and thepercentage of progress made. An interesting open problem is understanding the severity of the locomotionerror and deciding on whether to use a recovery technique. The robot may also have to decide whichrecovery technique to use.

2.3 Dynamics Study Using Affine Connections

We will formulate the kinematic reduction problem using affine connections, and this section provides theimportant definitions that we need. For those unfamiliar with affine and Riemannian differential geometry,we suggest Abraham and Marsden [1] and Kobayashi and Nomizu [34].

Lewis and Murray [20] provide a foundation for studying simple mechanical systems whose Lagrangianis the system kinetic energy minus the potential energy. In this subsection, we will consider only systemswith constant potential energy to explain the concepts. The equations of motion for a mechanical system SD

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Figure 10: A single degree of freedom hopping robot that self-orients itself before propulsion (Courtesy:Caltech/JPL).

with an n-dimensional configuration space Q and a Riemannian metric g : T Q→ R can be expressed as

∇ c(t )c(t) = ua(t)Ya(c(t)), (5)

where c(t) = (q1(t), . . . ,qn(t)) is a curve on Q, ∇ is an arbitrary affine connection defined by

∇XY = (∂Y i

∂q j X j + ΓijkX jY k)

∂∂qi (6)

for a set of Christoffel symbols Γijk, ua ∈R are the inputs, Y1, . . . ,Ym are the control vector fields, X ∈ TqQ is

a vector field, and we use the Einstein summation notation, where we sum over all possible values of indiceswhich occur twice in a single term. For example,

ua(t)Ya(c(t)) =m

∑a=1

ua(t)Ya(c(t)). (7)

Some important properties that will help characterize dynamic systems are accessibility and controllabil-ity. We will now discuss each property using a function called the symmetric product.

Definition 2.1 [20] Given an affine connection ∇ on Q, the symmetric product is defined as

〈X : Y 〉= ∇XY + ∇Y X , ∀X ,Y ∈ T Q.

An important property for control systems is the notion of local configuration accessibility, the ability toreach an open set of configurations starting from rest. Lewis and Murray [20] give the sufficient conditionfor local configuration accessibility using the symmetric product for SD. Let Ddyn = Span{Y1, . . . ,Ym} be the

input distribution for SD, Sym(Ddyn) be the symmetric closure of Ddyn ⊂ T Q, and Lie(Ddyn) be the involutiveclosure of Ddyn.

Theorem 2.1 [20] The mechanical system SD is locally configuration accessible at q if

Lie(Sym(Ddyn))q = TqQ. (8)

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Using Theorem 2.1, Lewis and Murray show that a rigid body in the plane is locally configuration acces-sible with two inputs; one of the inputs is a force, and the other can be an off-centered force or a torque.Thus, with these two inputs, the rigid body can reach an open set of local configurations starting from rest.

Before we state the sufficient condition for Small Time Local Configuration Controllability (STLCC) ofmechanical systems, the ability to reach a local neighborhood of configurations starting from rest, we definethe notion of good and bad symmetric products. A symmetric product is bad if it contains an even numberof each of the vector fields Ya,a = 1, . . . ,m; otherwise a symmetric product is good. Lewis [19] gives thesufficient condition for STLCC:

Theorem 2.2 [19] If Dim(Lie(Sym(Ddyn))q) = n, and every bad symmetric product at q ∈Q is an R-linearcombination of good products of lower degree at q, then SD is STLCC at q.

Lewis [19] proves configuration controllability for the snakeboard [17], a variation of the skateboardwhere the wheel directions can be changed and there is a rotor; that is, with input torques for the wheel-directions and rotor, the snakeboard can reach a local neighborhood of configurations starting from rest.

Configuration controllability is useful, but planning for dynamic models of mechanical systems is stilldifficult compared to planning for kinematic models. What is a kinematic model?

Definition 2.2 A driftless kinematic model SK with an n-dimensional configuration space Q, is defined bythe set of linearly independent vector fields {X1, . . . ,X m} such that

q(t) = uα(t)Xα(c(t)), (9)

where q ∈ Q and uα ∈ R.

Bullo et. al [9] look at conditions when a mechanical system can be represented by a kinematic model. Let(c(t), u) be a solution of (9) for controls u ∈ Ukin, and Dkin = Span{X1, . . . ,X m} be the input distributionfor the kinematic model. The kinematic model SK is said to be a kinematic reduction of SD if there existcontrols u ∈Udyn so that (c(t),u) is a solution of (5). Note that Udyn is not the same as Ukin. The relationshipbetween Dkin and Ddyn is summarized as:

Theorem 2.3 [9] The kinematic model SK is a kinematic reduction of SD if and only if the distributiongenerated by the vector fields {Xi,〈X j : Xk〉|i, j,k ∈ {1, . . . , m}} is a constant rank distribution contained ininput distribution Ddyn.

In addition to the condition placed by Theorem 2.3 on Ddyn and Dkin, Ukin must be such that there exists u ∈Udyn that can track the trajectory; for example, Ukin must at least be C2 continuous. Bullo and Lynch [10] findkinematic reductions to a three-degree-of-freedom manipulator with a passive joint by finding decouplingvector fields (a decoupling vector field is a vector field that can be tracked at arbitrary time scaling andthat satisfies the constraints on the system). This is significant because we can plan paths along decouplingvector fields ignoring effects like inertia and drift. Bullo and Lynch also present the underwater vehicleexample where they choose the dynamic input vector fields to be the decoupling vector fields, that is, theyset Xi = Yi. Lewis [18] shows that the upright rolling disk is kinematically reducible; that is, we can ignorethe dynamic effects of inertia while planning paths for the upright rolling disk. This greatly simplifies theplanning problem; if we find smooth plans for the kinematic model, those trajectories can be tracked by thedynamic system.

Typically, it is not straightforward to find a kinematic model for a mechanical system. For a genericmechanical system to admit a kinematic reduction, the system must satisfy certain properties, as expressedin Theorem 2.3. Bullo and Lynch [10] provide a direct algorithm for finding decoupling vector fields usingthe conditions that they must satisfy: the dynamic constraints on the system at arbitrary time scaling. An

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important property of the Yaw model that will help us derive its kinematic reduction is that its Lagrangianis invariant to yaw rotations, that is, there is a symmetry. For the Yaw model, by Noether’s theorem [8], thismeans that the momentum about the yaw axis is conserved in the absence of external forces. In Section 5.4.3,we will use this symmetry to derive the kinematic reduction for the Yaw model.

3 Problem Statement

The following is the problem statement for this thesis:

1. We propose a detailed analysis of a new locomotion technique for legged robots, legless locomotion,that couples the external nonholonomic constraints and the body rotations produced by the dynamicsof swinging legs. The analysis will involve finding simplified, approximate, and generic modelsfor legless locomotion. The approximate models will help study the specific influence of the systemdynamics and kinematics on the overall motion. The generic model will encapsulate the key propertiesof legless locomotion.

2. We will develop a classification structure for locomotion errors, which will focus on errors arisingdue to the environment or situation. We propose legless locomotion as a recovery technique for thecase when a legged robot is high-centered and will evaluate it based on criteria like speed and energyconsumed. Finally, we will place legless locomotion in the context of existing locomotion techniques.

3. We will find kinematic reductions for legless locomotion using the system symmetries. This will helpunderstand the phenomena behind legless motion and assist planning by finding drift-free representa-tions. Given the complexity of the system, we may not succeed in finding complete reductions, butmay find reductions for some body freedoms.

4 Technical Approach

This Section defines the approach we will take to address the questions raised by the problem statement.The technical approach is organized in three parts:

• Understanding legless locomotion using simple models.

• Classification of locomotion, errors, and recovery.

• Finding kinematic reductions.

4.1 Understanding Legless Locomotion

Our approach to understanding legless locomotion will include two primary elements:

• Finding simplified models.

• Finding a generic model.

Finding Simplified Models

The key approach to understanding a mechanical system will be to find simplified models that capture itsessence. We will then use the simplified models to understand useful properties for the original system. Wewill use three types of simplifications:

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Figure 11: A planar eccentric-mass wheel performs harmonic oscillations for small amplitude.

Figure 12: The simple pendulum performs harmonic oscillations for small amplitude.

1. Decoupling the system’s internal dynamics and external contact kinematics.

2. Reducing the number of degrees of freedom of the system, that is, restricting the robot’s configurationspace.

3. Finding kinematic reductions for the dynamic system.

The first type of simplification, decoupling the system dynamics and external contact kinematics, helpsunderstand the individual influence of the dynamics and the kinematics on the overall robot motion. It alsotells us the influence of the dynamics on the kinematics and vice versa. For example, an eccentric masswheel (see Fig. 11) in the plane has both kinematics of contact and dynamics produced by the stabilizinggravitational force. Let us now consider splitting the dynamics from the kinematics. If we keep bodyrotations small in the eccentric-mass wheel, then its geometric center does not move much; we can thenapproximate the motion by neglecting the influence of translation on the rotational dynamics. Thus, wepivot the body at its geometric center and then study the body rotational dynamics as though it is a pendulum(see Fig. 12). Once we know how the body oscillates, we can estimate the motion of the wheel in the planeusing the contact kinematics. The second model belongs to a class that we call Pivoting Dynamics (seeSection 5.2.2 for more details) and is simpler, because the kinematics equations are first-order and thedynamics and kinematics are decoupled. Clearly, there will be differences between the simplified model andthe original model, but if the results are similar, then using the simplified model is useful.

The second type of simplification, restricting the system’s degrees of freedom, is easier to implement; wewill merely reduce the robot’s configuration space by disabling some joints. This reduction in the body’sfreedoms will help understand the relationship between the different robot freedoms; in particular, thistechnique will help us understand the relationship between the actuated freedoms and the passive freedoms.For example, consider a satellite in R3 with three reaction wheels with perpendicular axes. We can restrictsome of the satellite’s rotational freedoms to understand the influence of the reaction wheel motions on theremaining freedoms. This way we can understand the influence of controls on the various passive freedoms.

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The third type of simplification, finding kinematic reductions, is more involved. It involves identifying ifthe dynamic system with acceleration inputs and drift can be thought of as a driftless kinematic system withvelocity inputs, at least for some degrees of freedom. This requires first developing a kinematic model andthen proving the properties of Theorem 2.3. See Section 4.3 for more details.

Once we get the simplified models, we can then understand what insight each simplified model providesinto the original mechanical system. For example, in Section 5, we consider single-axis models like theeccentric-mass wheel and the simple pendulum to understand the influence of mass distribution and contactconstraints on the oscillatory frequencies of RRRobot and the Pivoting Dynamics models. These simplifiedmodels help us understand the oscillatory frequencies noticed in RRRobot and the Pivoting Dynamics mod-els. We will also explore the validity of the simplified models; for example, why does the Pivoting Dynamicsmodel move faster than the RRRobot simulations? See Section 5.2.2 for more details.

Legless Locomotion Generic Model

RRRobot is good to demonstrate legless locomotion, but it seems that there are still many variables in itsdesign; for example, RRRobot has two legs with aligned rotational axes. Why should the axes be aligned?What about orthogonal leg axes? How do we choose gaits for RRRobot? What should the phase relationshipbetween the legs be? There are many similar questions. This thesis will find a simple generic model forlegless locomotion. This is similar to the idea of finding templates for insect and animal locomotion. Fulland Koditschek [28] define a template as a model created by trimming away all the incidental complexity ofjoints and that which serves as a guide for locomotion control. In short, templates are useful to understandthe broad principles underlying a physical system. What are the broad principles of legless locomotion asrepresented by RRRobot?

• Underactuation: RRRobot has seven degrees of freedom with just two actuators.

• Out-of-phase body oscillations: this is an important feature of the body rotations that produce loco-motion. For example, during lateral translation, the body performs out-of-phase pitch-yaw rotations;during circular translation, the body performs out-of-phase roll-pitch-yaw rotations.

• Exploiting inertia differences: RRRobot produces net yaw using out-of-phase leg motions and ex-ploiting the varying system inertia.

Fig. 13 shows a simple generic model we have in mind for legless locomotion. The template has only oneleg, pivoted at the geometric center of the sphere on a virtual joint. The leg is massless but has a pointmass at its distal end. What are the freedoms for the leg? The leg is connected to the sphere at a sphericaljoint; that is, the leg has three rotational degrees of freedom. We will assume that all the three freedoms areactuated.

We believe that this generic model captures the important elements of legless locomotion. Clearly, thegeneric model is underactuated (eight degrees of freedom with just three actuators); out-of-phase bodyrotations can be produced using, say, circular leg motions; inertia differences can be created using suitableleg roll, yaw, and pitch configurations. Table 2 shows some examples on how this generic model can actuatebody rotations by using small leg motions. In fact, using a single leg pivoted at a spherical joint, the robot hassome capabilities beyond the two-legged RRRobot; for example, the model can tilt itself while locomotingby moving the leg to either side. There are some restrictions on body motions though; for example, bodyyaw is impossible when the leg is vertical, since the leg has no inertia; but body yaw is possible by rollingor pitching the leg slightly, and then yawing the leg. It will be interesting to see if this generic model canachieve all the body rotations required to produce locomotion. We will study this problem using modelingand simulations.

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Figure 13: Legless locomotion generic model.

Body Rotation Leg rotation

Pitch YYaw Two ways:

1) Y or X; then Z.2) Y or X leg from vertical position; +Z;Y or X back to vertical position; -Z.

Roll XPitch and Yaw Y and ZRoll and Yaw X and Z

Roll, Pitch, and Yaw X, Y, and Z

Table 2: Achieving body rotations in the legless locomotion generic model (see Fig. 13) using leg rotations.

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If we want to compare the generic model’s motion with the two-legged RRRobot, the challenge will beto find equivalent leg motions for the model that achieve the RRRobot body rotations. If we can find theseequivalent leg motions for the generic model, then the two legs in RRRobot effectively reduce to a singleleg. So, how do we measure this equivalence? We could compare the model’s motion with the two leggedRRRobot by two means:

• Compare time-histories of the body rotational trajectories: this is better than comparing net rotationalmotion, because in the presence of gravity, pitch and roll configurations are always changing; hence,net rotational motions are difficult to define for RRRobot.

• Gross planar motion: net planar translation is the main goal of locomotion, and we will comparethe gross distances and directions travelled in both cases. We will also compare the average pathsfollowed by the two systems.

Understanding legless locomotion using a multi-degree of freedom single leg is difficult, because of thecomplex interaction between the dynamics and kinematics. To study just the interaction between leg motionsand body motions, we will approximate RRRobot on a plane by pivoting the robot on a spherical joint at thehemisphere center and ignore the effects of translation on the body dynamics (Pivoting Dynamics). Oncewe compute how the body rotates for a given leg motion, we can approximate the translation in the planeusing the contact kinematics. Thus, we use an approximate simplified model where the kinematics anddynamics are decoupled. To study the influence of leg motion on the body attitude dynamics, we will usethe techniques of Walsh and Sastry [37] and Rui et al. [30]. Walsh and Sastry discuss a method of controllingthree linked bodies in SE(3) with two spherical joints, and Rui et al. provide a technique for attitude controlof a satellite in SO(3) with multiple appendages and reaction wheels.

4.2 Classification of Locomotion, Errors, and Recovery

In this section, we will first place legless locomotion in the context of the properties of existing locomotiontechniques. We will then present a classification structure for mobile robot locomotion errors and, finally,project legless locomotion as a recovery technique for high-centeredness.

Locomotion Classification

There is a wide range of locomotion techniques available, and it is useful to have a classification of thevarious locomotion techniques. Yim’s locomotion classification structure [38] focusses only on static lo-comotion (see Fig. 14). We will use three hierarchical levels to classify the dynamically stable locomotiondomain and incorporate legless locomotion into the structure. See Fig. 15 for a first pass at the problem.

• Level 1 discusses the nature of dynamic stability, whether the system is asymptotically stable, neu-trally stable, or unstable [29]. A stable system, after a perturbation, returns to a stable limit cycle orequilibrium point after a perturbation. An asymptotically stable system returns to the original limitcycle or equilibrium point, while a neutrally stable system returns to a new nearby limit cycle orequilibrium point.

• Level 2 discusses the nature of contact between the robot and environment, depending on whetherthere is slip between bodies. The Universal Planar Manipulator [27] requires slip between objects andthe table surface on which the parts are moved around. Legless locomotion requires no slip betweenthe round body and the surface.

• Level 3 discusses the nature of dynamics underlying the locomotion technique. The nature of dy-namics depends on whether there is a flight phase, in which case the dynamics is hybrid. Also, the

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Figure 14: Statically stable locomotion classification [38].

Figure 15: Dynamically stable locomotion classification: there are three levels, and at each level, we checkif the locomotion technique satisfies the property.

dynamics depends on the nature of forces used to propel the robot; ‘normal’ legged locomotion canhave flight phases (for example, running) and has directly actuated appendages. Legless locomotiondoes not have a flight phase and uses reaction masses to propel the body.

Legless Locomotion is dynamic, asymptotically stable (in the presence of small disturbances and externaldamping), and requires pure rolling contact between its body and the environment. It has no flight phaseand uses reaction masses to produce forces.

Locomotion Error Classification

For making robotic locomotion more robust in complex terrain, it is essential to have an understanding ofhow robots get trapped. To date, there has not been a systematic analysis of locomotion errors. Before weattempt to classify locomotion errors, we need definitions for locomotion modes and errors.

Definition 4.1 A locomotion mode is a node in the locomotion classification structure (see Figs. 14 and 15).

A locomotion mode influences the nature of contacts and dynamics and can include many types of loco-motion gaits or leg motion patterns. For example, running and trotting are gaits that have the followingcommon characteristics: neutral/asymptotic stability, point contact, direct actuation, and a flight phase, but

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Figure 16: Locomotion error

Category ExamplePresence of an undesired contact Bipedal locomotion with the

body touching the groundAbsence of desired contact High-centered robotsPresence of a kinematic singularity Loss of body degree of freedom

when all legs are fully extended.

Table 3: Examples of situational errors (see Fig. 17).

their leg-ground contacts are arranged differently during each leg cycle. Thus, they will be included in dif-ferent locomotion modes. Similarly, a walking gait for a multi-legged robot with some legs disabled belongsto a locomotion mode different from the walking gait with all legs functioning properly.

Definition 4.2 A locomotion error is an undesired external event that forces a change from one locomotionmode to another mode (see Fig. 16).

For example, an animal changing from the trot mode to the running mode of its own volition is not alocomotion error; but a multi-legged robot breaking some legs will constitute an error, since this will forcethe robot to use a different locomotion mode.

Using Definitions 4.1 and 4.2, we can classify locomotion errors. Fig. 17 shows a first pass at the problem.

• Level 1 categorizes errors based on whether the cause of the failure is in software or hardware orwhether the error is due to a peculiar situation that the robot is in.

• Level 2: Within situational errors, the failure may be due to the presence of an undesired contact,the absence of a desired contact, or the presence of a kinematic singularity. Table 3 illustrates eachcategory with examples.

• Level 3: Absence of a desired contact can occur in two ways: either there is no contact at all (as inthe high-centered case), or there is intermittent contact. Intermittent contact can occur in a varietyof ways; for example, slipping on a partially greased floor, or partially high-centered robot that hasleg-ground contact only at certain body configurations.

This thesis will focus on locomotion errors due to situations that result in the absence of a desired contact(no contact).

Locomotion Error Recovery

We propose legless locomotion as a technique for robot error recovery from a high-centered state. Our strat-egy will be to use legless locomotion until the legs touch the ground, after which conventional locomotioncan be used. We will study through experiment how effective legless locomotion is as a technique for robotsto escape by comparing sliding and rolling legless locomotion (see Section 1 for more details). In the for-mer case, RHex will have a flat bottom, and in the latter, a round bottom will be strapped onto RHex. Themetrics we will use will be speed and energy; that is, for both types of legless locomotion, we will measurethe speed and how much energy is consumed in moving in the plane from one point to another. This willhelp us understand if a round bottom is better than a flat bottom for legless locomotion.

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Level 1

Level 2

Level 3

Figure 17: Locomotion error classification

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4.3 Finding Kinematic Reductions

A kinematic model has first order differential equations and planning is much simpler. Apart from inte-grating the dynamic equations of motion, a tough process, there exists no systematic procedure for findinga kinematic representation for a dynamic system. One way of finding kinematic reductions is to look forsystem invariances or symmetries; this can be deduced by noticing that the system’s Lagrangian is invariantto translations and rotations on the configuration space, that is, the Lagrangian does not vary with changes inthe system configuration with respect to a spatial frame (a frame fixed in space. See [24] for more details).By Noether’s theorem, this system symmetry means that there is a conserved momentum map, and this givesa first order differential equation relating velocities and configurations. Section 5 shows this procedure for asimplified model, the yaw model. It’s not possible to find complete kinematic reductions to all mechanicalsystems, but we may be able to find kinematic reductions to at least some of the system’s freedoms. Thiswill depend on the number of decoupling vector fields we can find and the function space for the kinematiccontrols. We hope to make progress on this problem by first finding kinematic reductions for the simplifiedmodels and then proceeding to the RRRobot model.

5 Work To Date

This section discusses the work done to date in relation to the questions raised by the problem statement.We have performed some informal experiments with RHex that provides motivation for understanding loco-motion errors. We have found some legless locomotion models and demonstrated correspondence betweenexperiments and models. We have also illustrated the kinematic reduction ideas using the Yaw model.

5.1 Experiments with RHex

We performed several informal experiments in Summer 2002 to investigate how RHex can get stuck. Thestructured environment consisted of various arrangements using cinder-blocks and styrofoam. The keyelements we focussed on were what happens when RHex’s body or legs are trapped. It turned out that RHexwas able to escape in most circumstances by sheer strength, except when RHex gets high-centered. In thehigh-centered case, we had to try many different leg-motion patterns to excite rocking body-motions andincrementally translate the robot. This gave us the motivation to understand the problem of locomotinghigh-centered robots. See http://www.cs.cmu.edu/∼bravi/research.html for the movies.

5.2 Legless Locomotion Models

5.2.1 RRRobot on a plane

We begin studying legless locomotion by exploring RRRobot on a plane. The RRRobot-on-a-plane modelis a hemispherical shell with two short actuated legs (see Fig. 18). The massless shell has radius r, and themassless legs have length l. There are five masses on the robot indicated by black dots: a mass at the distalend of each leg (Ml), a mass where each leg is pinned (Ms), and a mass at the bottom of the shell (Mb).Torques τ1 and τ2 may be applied at the leg joints, and the shell rolls on the plane without slip.

The configuration q of RRRobot on a plane consists of the sphere’s position and orientation (x,y,R)with respect to a spatial frame and the internal configuration of its legs (φ1,φ2). Here R = R(θ1,θ2,θ3) ∈SO(3) represents the orientation of the sphere according to the ZXY fixed-angle convention [12]. Thus,q = (x,y,R(θ1,θ2,θ3),φ1,φ2)T ∈ R2×SO(3)×R2.

The equations of motion for RRRobot on a plane take the form

M(q)q +C(q, q)q + G(q) = τ + (λ1ω1)T + (λ2ω2)T , (10)

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Figure 18: RRRobot on a plane.

Figure 19: The Pivoting Dynamics model simplifies the RRRobot-on-a-plane model (see Figure 18) intotwo parts: (a) RRRobot pivoted at its geometric center on a spherical joint and (b) a sphere on a plane.

where M(q) ∈ R7×7 is the positive-definite non-diagonal variable mass matrix, C(q, q)q ∈ R7 is the vectorof Coriolis and centrifugal terms, G(q) ∈ R7 is the vector of gravitational terms, τ = (0,0,0,0,0,τ1,τ2)T isthe generalized force, and ωa ∈R7,a = 1,2, are the rolling constraints for a sphere-plane contact [23] givenby

ω1 = (1,0,0,r sinθ1,−r,0,0),

ω2 = (0,1,r cosθ3,r cosθ1 sinθ3,0,0,0), (11)

and λa ∈R is the magnitude of the contact constraint force. The input torques indicate that only the legs areactuated.

5.2.2 Pivoting Dynamics Model

The RRRobot-on-a-plane model includes the interplay between body dynamics and contact kinematics. Toanalyze just the interaction between leg motion and body attitude, we simplify the RRRobot-on-a-planemodel by pivoting the robot on a spherical joint and ignoring the effect of translation on body attitude dy-namics. Once we compute the body attitude motion for a certain leg trajectory, we use the contact kinematicsequations to approximately predict RRRobot translation in the plane. Thus, this model, called the PivotingDynamics model, approximately reduces the RRRobot system into two parts (see Fig. 19): 1) The dynamicsof RRRobot rotating about a spherical joint, 2) The contact kinematics of a sphere on the plane.

The configuration of the Pivoting Dynamics model qp consists of the sphere’s orientation R(θ1,θ2,θ3)with respect to a spatial frame and the configuration of its legs (φ1,φ2). Thus, qp = (R(θ1,θ2,θ3),φ1,φ2)T ∈SO(3)×R2.

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Figure 20: A planar eccentric-mass wheel performs harmonic oscillations for small amplitude.

The equations of motion for the Pivoting Dynamics model take the form

MPD(qp)qp +C(qp, qp)qp + G(qp) = τ, (12)

where MPD(qp)∈R5×5 is the positive-definite non-diagonal variable mass matrix, C(qp, qp)∈R5 is the vec-tor of Coriolis and centrifugal terms, G(qp)∈R5 is the vector of gravitational terms, and τ = (0,0,0,τ1,τ2)T

is the generalized force. The input torques indicate that only the legs are actuated and that there are no con-straints on the system. Once we compute the changes in body configuration for a certain leg trajectory, weuse the kinematic contact equations

(ω1

ω2

)q =

(00

)(13)

to compute the velocity of the contact point in the plane, where ω a is given in (11), and q = (x,y,qp).We simulated this model and compared the results with the RRRobot simulation (see Section 5.3 for moredetails).

5.2.3 Single-Axis-Rotation Models

If we consider body attitude changes only about one axis, say, the X or Y axis, then RRRobot on a planeis similar to a planar wheel with an eccentric mass (see Fig. 20); we call these the Single-Axis-Rotationmodels. The location of the mass and the inertia of the system is determined by the weight distribution onthe robot. If r is the wheel radius, M is the lumped mass of the system, and ρ is the radius of gyrationof the system with respect to an axis passing through the contact point and perpendicular to the plane, thetime-period for small amplitudes is

Tw = 2π

√ρ2

g(r−ρ), (14)

where g is gravity. Note that Tw decreases as r increases, and Tw increases as ρ increases.If the Pivoting Dynamics model is restricted to oscillate about the X or Y axis, then the Pivoting Dynamics

model is similar to a simple pendulum (see Fig. 21), whose time-period is

Tsp = 2π√

ρg, (15)

where ρ is the radius of gyration, and g is gravity. The time-period Tsp increases as ρ increases.Note that to get similar oscillatory behavior between the eccentric mass wheel and the simple pendulum,

a specific mass-distribution may be required; for example, for the simple pendulum, if ρ = r/2, then we get

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Figure 21: The simple pendulum performs harmonic oscillations for small amplitude.

Table 4: Rotation Time-Periods for the RRRobot-on-a-plane model and the Pivoting Dynamics model

X Rotations (sec) Y Rotations (sec)RRRobot-on-a-plane 1.29 1.07Pivoting Dynamics 1.19 0.96

equal oscillatory time-periods. Table 4 shows the time-periods for roll and pitch rotations for the RRRobot-on-a-plane model and the Pivoting Dynamics model. The time periods of the two models are close to eachother, because ρ = r/2 for the pivoting dynamics model.

5.3 Producing RRRobot Translation

There are two oscillators in the RRRobot system due to gravity– the X attitude oscillator and the Y attitudeoscillator. It seems natural to use sinusoidal leg trajectories or gaits to control the body oscillatory motion.We present three sinusoidal gaits of the form asin(ωt +β )+γ that produce RRRobot translation. In all threegaits, the legs are π/2 out-of-phase with each other. The legs oscillate about the vertical position in Gait 1,the horizontal position in Gait 2, and π/4 off the horizontal in Gait 3 (see Table 5 for the gait parameters). Weused the following intuition to choose the gait parameters. When the legs are π out-of-phase, they produceonly roll if the home position is horizontal, or they produce yaw motion if the home position vertical. Whenthe legs are in-phase, they produce only pitch motion. Since we want a combination of body rotations, itseems logical to try π/2 out-of-phase gaits. We ran simulations of the RRRobot-on-a-plane model and thePivoting Dynamics model for fifty seconds, while we ran the robot experiment for one hundred seconds.The robot starts from rest at the origin with the legs in the vertical position, and a proportional-derivativecontroller is used to track the position trajectories. We use Ms = 0.053 kg, Ml = 0.057 kg, Mb = 0.3 kg,r = 0.12 m, l = 0.1 m, g = 9.81 m/s, and a damping coefficient k = −0.01. In the experiments, a verticaltether provides the servo power and control signals. We keep body oscillations small to minimize externaldisturbances from the tether. In simulation and experiment, we observe initial transients in the robot motion.

Gait 1

Gait 1 produces sideways translation (see Fig. 22) due to pitch-yaw body attitude oscillations, while rollbody attitude changes are negligible.

Gait 2

Gait 2 produces counter-clockwise circular translation (see Fig. 23) due to a combination of roll-pitch-yawbody attitude oscillations. The robot completes a circle in the RRRobot-on-a-plane simulation, completes

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Table 5: Gait parameters

Gait 1Leg 1 Leg 2

Amplitude a (rad) 0.3 0.3Frequency ω (rad/s) 8 8

Phase β (rad) 0 π/2Offset angle γ (rad) π/2 π/2

Gait 2Leg 1 Leg 2

Amplitude a (rad) 0.3 0.3Frequency ω (rad/s) 8 8

Phase β (rad) 0 π/2Offset angle γ (rad) π/4 π/4

Gait 3Leg 1 Leg 2

Amplitude a (rad) 0.3 0.3Frequency ω (rad/s) 8 8

Phase β (rad) 0 π/2Offset angle γ (rad) 0 0

one and a half circles in the Pivoting Dynamics Model simulation, and almost completes a half circle inexperiment.

Gait 3

Gait 3 produces backward translation (see Fig. 24) due to roll-yaw body attitude oscillations, while pitchbody attitude changes are negligible. There is a large roll translation at the start, because the legs movequickly from the vertical position to sinusoidal oscillations about the horizontal. Gait 3 does not producemuch translation, because the roll-yaw oscillations are small and surface stickiness possibly restricts motion.Our experience indicates that this gait is the least reliable of the gaits explored so far. In all three gaits,swapping the relative phase between the two legs produces translation in the opposite direction.

The paths followed by the contact point in simulation and experiment match well, but there is one cleardifference- the robot in experiment moves slower than in simulation. This may be due to unmodelled surfacefriction, slip between the body and the surface, or a deformed spherical shape at the contact point.

The translation produced in the Pivoting Dynamics model and in the RRRobot-on-a-plane model matchwell; the contact point follows similar paths, but the Pivoting Dynamics model moves faster, especially forGaits 1 and 3. This is because the Pivoting Dynamics Model is pivoted at its geometric center, while in theRRRobot-on-a-plane Model, the robot has a rolling contact. Thus, for a given change in attitude, the point ofcontact moves faster in the Pivoting Dynamics model than in the RRRobot-in-a-plane Model. In summary,if we are interested in gross planar translation, we can use the Pivoting Dynamics model to approximateRRRobot motion.

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Figure 22: Planar plots of contact point time history during sideways locomotion produced by Gait 1 in(a) RRRobot-on-a-plane simulation, (b) RRRobot-on-a-plane experiment, (c) Pivoting Dynamics modelsimulation. The solid arrow gives robot motion direction, and the dotted lines indicate the robot position atthe specified time.

5.4 Kinematic Reduction for the Yaw Model

5.4.1 Configuration Accessibility

We now refer back to the Yaw model introduced in Section 1.2. In particular, note that Y1 = g−1

010

, and

Y2 = g−1

001

. Since 〈Y1 : Y1〉, 〈Y2 : Y2〉, 〈Y1 : Y2〉 ∈Ddyn, Sym(Ddyn) = Ddyn. Except at sinφ1 = sinφ2 = 0,

Rank({Y1,Y2, [Y1, [Y1,Y2]]}) = 3. Thus, by Theorem 2.1, Lie(Sym(Ddyn)) = TqQ, and the Yaw model islocally configuration accessible everywhere except when both legs are horizontal. From now on, we willconsider only non-singular configurations. Thus, the Yaw model can reach an open set of configurationsstarting from rest.

5.4.2 Configuration Controllability

In 5.4.1, we showed that the Yaw model is configuration accessible. The bad symmetric products 〈Y1 : Y1〉and 〈Y2 : Y2〉 are in the span of Ddyn. Thus, by Theorem 2.2, the Yaw model is STLCC, and can reach a localneighborhood of the initial configuration starting from rest.

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Figure 23: Planar plots of contact point time history during forwards locomotion produced by Gait 2in (a) RRRobot-on-a-plane simulation, (b) RRRobot-on-a-plane experiment, (c) Pivoting Dynamics modelsimulation. The solid arrow gives robot motion direction, and the dotted lines indicate the robot position atthe specified time.

5.4.3 Kinematic Representation

Using (3), we now derive a kinematic representation of the Yaw model. Note that row 1 of (3) is integrable;that is, if the initial system velocity is zero, we can integrate row 1 to get

g11θ + g12φ1 + g13φ2 = 0. (16)

The terms on the left side of (16) add up to the system’s yaw momentum; thus, the yaw momentum is alwayszero.

Using (16), we define the kinematic representation of the Yaw model to be

q = X1u1 + X2u2, (17)

where X1 =

−g12/g11

10

, X2 =

−g13/g11

01

, Dkin = Span{X1,X2}, ua ∈ R, and we have assumed

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Figure 24: Planar plots of contact point time history during forwards locomotion produced by Gait 3in (a) RRRobot-on-a-plane simulation, (b) RRRobot-on-a-plane experiment, (c) Pivoting Dynamics modelsimulation. The solid arrow gives robot motion direction, and the dotted lines indicate the robot position atthe specified time.

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Figure 25: Body yaw when leg 2 is fixed and leg 1 is rotated.

that we have direct control over the leg joint velocities. This kinematic representation is also invariant toyaw rotations. Comparing (17) and (3), we note that Xi 6= Yi (unlike the underwater vehicle example in [10]).

5.4.4 Kinematic Reducibility

Using linear algebra, we can verify that the vector fields X1 and X2 are in the span of Ddyn. Also, thesymmetric products ∇Xi

X j, i, j = 1,2 are in the span of Ddyn. Thus, by Theorem 2.3, the Yaw model is

reducible to the kinematic representation in (17), while ensuring C2 continuous inputs for the kinematicrepresentation. Note that (17) is much simpler than (3), but represents all the properties of (3). We have,thus, reduced planning for the mechanical system in (3) to a nonholonomic kinematic path planning problemfor (17).

Further, we notice that if only one leg moves and the other is fixed, the angular momentum equation (16)can be integrated again. This induces a holonomic constraint on the Yaw model; that is, the yaw configura-tion is specified just by the leg configuration and is independent of the path taken by the leg. Suppose leg 2is kept fixed at φ2, and leg 1 is moved by ∆φ1 from π/2, the net body yaw is calculated using the holonomicconstraint

∆θ =2mllb arctanh(a(∆φ1))√

mll2γ(φ2)

, (18)

where γ(φ2) = (4mll2−2(4mlb

2 +3mll2 +4mmb2 +mll

2 cos2φ2)), and a(∆φ1) =−2√

ml l2 sin(∆φ1)/γ(φ2).

Thus, when leg 1 is moved to φ1 = 0, the body rotates by an angle ∆θ =2ml lb arctanh(2

√ml l

2/γ(φ2))√ml l

2γ(φ2). Fig. 25

shows how the body yaws, as a function of leg 2 configuration, when leg 1 is moved. Since the yaw inertiais largest when leg 2 is kept horizontal, the net body yaw is least. Also, there is maximum yaw, when leg 1is at the horizontal positions. Similarly, if leg 1 is kept fixed at φ1, and leg 2 is moved by ∆φ2 from π/2, thenet body yaw is computed using the holonomic constraint:

∆θ =−2mllb arctanh(a(∆φ2))√mll

2γ(φ1), (19)

where γ(φ1) = (4mll2−2(4mlb

2 +3mll2 +4mmb2 +mll

2 cos2φ1)), and a(∆φ2) =−2√

ml l2 sin(∆φ2)/γ(φ1).

5.5 Yaw Model Motion Planning

The Yaw model kinematic reduction in 5.4.3 is simple to plan paths for, since we require (for configurationcontrollability) the following two leg-motion patterns or gaits only, one a sinusoidal trajectory and the othera cubic spline trajectory.

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• Gait A: Move the legs simultaneously with out-of-phase velocities; for example, φ1 = Asin t, and φ2 =Acos t for some suitable amplitude A. To ensure that the dynamic inputs can track these velocities, anenvelope function is used at the start and end of each cycle to guarantee smoothness in velocity space(see Section 2.3 for conditions on the kinematic model input space).

• Gait B: Move only one leg, say, using a cubic spline with zero initial and final velocities; keep theother leg stationary.

Each cycle of Gait A produces net body yaw due to the varying mass matrix (see Section 1.2 for intuitivethought experiments on net yaw produced by interleaved leg motions). Gait B produces body yaw, but if theleg returns to the start configuration, then the net body Z rotation is zero (see 5.4.4 for more details).

Here is a motion planning algorithm for the Yaw model. Without loss of generality, we will assume thatthe legs start from the legs-up configuration. Let the goal configuration be qg = (θg,φ1g,φ2g).

1. Precompute body yaw α using (18) when moving leg 1 from π/2 to φ1g using Gait B while keepingleg 2 fixed at π/2. Similarly, precompute body yaw β using (19) when moving leg 2 from π/2 toφ2g using Gait B while keeping leg 1 fixed at φ1g. We can compute α and β a priori because theYaw model is invariant to yaw rotations.

2. Choose amplitude A so that the net yaw ρ in one cycle of Gait A is sufficiently small to make c =(θg− (α + β ))/ρ an integer. Repeat Gait A c times.

3. Move leg 1 to φ1g using Gait B. Move leg 2 to φ2g using Gait B.

Once we compute the trajectories in the kinematic representation for each leg to achieve qg, we can trackthose trajectories in the mechanical system using a simple proportional-derivative controller.

Yaw Model Planning Numerical Results

The start configuration qs = (0,π/2,π/2) is the legs-up configuration. Let mm = 1.0 kg, ml = 0.5 kg,b = 0.5 m, l = 0.5 m, A = 0.516, and qg = (−4π/180,4π/9,4π/9) (four degrees body yaw and ten degreesoff the legs-up configuration). For Gait A, we set the leg velocities as follows:

φ1 = f (t)Asin(t),

φ2 = f (t)Acos(t),

where

f (t) =

{1− e−kmod(t,2π) mod(t,2π)≤ π,1− e−k(2π−mod(t,2π)) otherwise,

(20)

(see Fig. 26) and k = 100. The envelope function ensures that the legs’s initial and final velocities duringeach cycle of Gait A are zero. Following the algorithm, we precompute α = −0.02889 and β = 0.02875.With ρ =−0.014, we repeat Gait B five times so that the body reaches the required yaw configuration. Now,when the legs are moved one by one to their goal positions using Gait B, the Yaw model will reach qg. Fig. 27shows the time history of body rotation and leg configuration when executing the plan on the mechanicalsystem using a proportional-derivative controller (constants Kp = 20 and KD = 10).

6 Schedule

Table 6 gives the proposed schedule for thesis research.

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Figure 26: Plots of the envelope function f (t) (see (20)).

Figure 27: Time history of body yaw and leg configurations while executing a plan to go from start config-uration qs = (0,π/2,π/2) to goal configuration qg = (−4π/180,4π/9,4π/9) (four degrees body yaw andten degrees off the legs-up configuration). Gait A is repeated five times in period (a), leg 1 is moved usingGait B into its goal position in period (b), and leg 2 is moved using Gait B into its goal position in period (c).

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Table 6: Schedule for proposed work.Task Duration (months)

Compare RRRobot models with experiment results 2– Understand differences in speed between model and experiment.– Compare mass distribution in the Pivoting Dynamics model and RRRobot.Develop legless locomotion generic model 2– Develop simulations.– Develop gaits.– Find gaits that match two-legged RRRobot.Evaluate legless locomotion on RHex 1.5– Experiments with legless locomotion and slip-related locomotion on RHex.Kinematic reductions for simplified models 2– Develop reductions for simplified models (Single-Axis and Pivoting Dynam-ics) in absence of gravity.– Explore mechanical systems with non-constant potential.Kinematic reductions for RRRobot yaw 1.5– Choose suitable coordinate system.– Develop invariant equations of motion.Kinematic reductions for RRRobot 2– Collect previous results.– Explore RRRobot in absence of gravity.Write and defend thesis 4

Total 15

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7 Contributions

Following the technical approach detailed in Section 4, this thesis will give us insight into mechanicalsystems. In particular, it will make contributions to the following areas.

• Understanding a dynamically coupled locomotion technique.

• Analyzing locomotion errors and recovery techniques.

• Finding kinematic reductions using symmetry.

We discuss each contribution below.

7.1 Legless locomotion

This thesis will present a detailed analysis of a new dynamically coupled locomotion technique for sys-tems with variable inertias, external constraints, and no contact between the actuated freedoms and theenvironment. We will discuss many approaches to developing simplified models to understand the originalmechanical system and understand the conditions for validity of the simplified models. For example, wewill compare results from the Pivoting Dynamics model and the RRRobot model. We will also present ageneric model that captures the essence of legless locomotion, develop gaits for it, and present a comparisonbetween the generic model and the two-legged RRRobot.

7.2 Mobile robot error classification and recovery for high-centered robots

Currently, there does not exist a systematic classification of mobile robot errors for locomotion, i.e., howrobots get trapped in the physical world. This thesis will focus on mobile robot locomotion errors that occurdue to interaction with the physical world. This thesis will also present legless locomotion as a specifictechnique when undesired contacts exist. We will investigate the efficacy of two strategies– slip-relatedtranslation and legless locomotion– for locomoting a high-centered robot. This will help us say if a flat-bodied or round-bodied is better for such situations. Finally, this thesis will place legless locomotion in thecontext of existing locomotion techniques.

7.3 Kinematic reductions for mechanical systems

Is there a systematic way of finding kinematic reductions using system symmetries for mechanical systems?Finding full reductions of mechanical systems may be difficult, but we could find kinematic reductionsfor specific degrees of freedom. This thesis will provide an approach to explore kinematic reductions forcomplicated systems by developing reductions for simplified models, starting with the simplified modelsand then expanding to the Pivoting Dynamics models. Finally, we will find a kinematic reduction for leglesslocomotion. This will assist in planning for legless locomotion.

8 Acknowledgment

This work was supported under NSF IIS 0082339, NSF IIS 0222875, and DARPA/ONR N00014-98-1-0747contracts. I thank my advisors Matt and Al for their insightful comments and patience. I also thank mycommittee members Francesco and Dimi for many useful comments. I thank Devin Balkcom for the manyfun discussions we have had and for reading this document. I thank Elie Shammas and Klaus Schmidt forthe great discussions on theoretical control. Sidd provided valuable comments on this document.

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