Passive Dynamic Object Locomotionby Rocking and Walking Manipulation

Abdullah Nazir and Jungwon Seo

Abstract— This paper presents a novel robotic manipulationtechnique for transporting objects on the ground in a passivedynamic, nonprehensile manner. The object is manipulated torock from side to side repeatedly; in the meantime, the forceof gravity enables the object to roll along a zigzag path thatis eventually heading forward. We call it rock-and-walk objectlocomotion. First, we examine the kinematics and dynamics ofthe rocking motion to understand how the states of the objectevolve. We then discuss how to control the robot to connectindividual rocking motions into a stable gait of the object. Ourrock-and-walk object transportation technique is implementedusing a conventional manipulator arm and a simple end-effector, interacting with the object in a nonprehensile mannerin favor of the passive dynamics of the object. A set ofexperiments demonstrates successful object locomotion.

I. INTRODUCTION

One interesting question in archaeology is how the giantrock statues of Easter Island (known as “moai”) were trans-ported several hundred years ago [1]. It is known that thereare more than 900 statues currently existing on the islandand some instances measure up to 10m in height and 74metric tons in weight. Historical evidences show that thelarge, heavy statues had been transported as far as 18kmdistance [1]. Fig. 1(a) (from [1]) describes one viable answerto the question. The figure shows a replica of the statue withthree ropes tied around the head, along with three groups ofpeople interacting with the statue through the ropes. Theyare demonstrating that it is possible to “walk” the statue byrepeatedly rocking it sideways1.

Motivated by the way the statue was manipulated inFig. 1(a), in this paper we are concerned with devising anovel robotic manipulation technique for object locomotionby rocking and walking. Fig. 1(b) briefly shows the progressof our manipulation process. The robot arm repeatedly rocksthe cone-shaped object sideways and the object can takea small step forward at each iteration. We first study thekinematics and dynamics of object rocking motion. We willthen investigate how to walk the object forward by alternatingrocking phases and taking advantage of gravity in between.Finally, our manipulation technique will be demonstratedwith a real robot system featuring the conventional manipu-lator arm as shown in Fig. 1(b).

Abdullah Nazir is with the Department of Electronic and ComputerEngineering, The Hong Kong University of Science and Technology, ClearWater Bay, Hong Kong sanazir@connect.ust.hk

Jungwon Seo is with the Departments of Mechanical and Aerospace/-Electronic and Computer Engineering, The Hong Kong University of Sci-ence and Technology, Clear Water Bay, Hong Kong junseo@ust.hk

1A video can be seen here: https://www.youtube.com/watch?v=J5YR0uqPAI8

(a)

(b)

Fig. 1. (a) Rocking and walking the 3m tall, 4.35 metric ton moai-replicawith teams of people handling three ropes. This figure is adapted from [1]with permission. (b) Our robotic rocking and walking technique (also seethe video attachment).

Leveraging the passive dynamics of an object of interest,our manipulation technique can be an energy-efficient wayto transport objects that are too large, heavy to apply thetraditional grasp-lift-and-carry approach. It can actually bethe only transportation option that the robot can choose at thetime. However, not all objects could be transported using thetechnique. There will be a certain class of objects for whichthe technique is suitable. The non-prehensile nature of ourmanipulation suggests that it is necessary to keep track ofthe state of the object being transported.

Following Sec. II on literature review, Sec. III formallydescribes our problem. In Sec. IV we study the kinematicsand dynamics of object rocking. Sec V addresses gait gen-eration and control. Implementation and experimentation ispresented in Sec. VI. We conclude in Sec. VII.

II. RELATED WORK

Our work is concerned with a manipulation technique thattakes advantage of the dynamics of the object to be handled.See [2] for a general introduction to dynamic manipulationtechniques. Some examples include quasidynamic tray tilting[3] and robotic juggling [4].

Considering the duality between manipulation and loco-motion, our work is also relevant to robotic locomotion.There is an extensive literature on robotic locomotion. See[5] and references therein. In particular, our object trans-portation technique takes advantage of the passive dynamicsof the object under gravity. Early works on passive dynamic

locomotion include the well-known passive dynamic walkingtechnique presented in [6]. [7] presents an efficient three-dimensional bipedal passive walker capable of knee flex-ion. [8] addresses the control aspects of a simple three-dimensional passive dynamic walker on a flat terrain. Adetailed empirical insight on human walking, in whichpassivity plays a key role, is presented in [9]. [10] exploitsthe passive falling dynamics of a modular loop robot, inwhich the robot continuously reconfigures itself to accelerateforward by adjusting the position of its center of mass.

Our object transportation technique is realized by ma-nipulating the object in a nonprehensile manner in whichthe object is not fully restrained by the end-effector. Earlyprogress in nonprehensile manipulation is summarized in[11] and its planning and control aspects are addressedin [12]. Notable examples of nonprehensile manipulationtechniques include pushing [13] and toppling [14]. Ourwork is similar to the pivoting technique [15], [16], butwe additionally leverage the passive dynamics of the object.Our previous works on picking through in-hand manipulation[17], robotic edge-rolling [18], and shallow-depth insertion[19] are also relevant to nonprehensile manipulation.

III. PROBLEM DESCRIPTION

The problem we address in the paper is concerned witha novel robotic manipulation technique that can be appliedto transporting an object of interest in a dynamic manner.Motivated by the way how the giant rock statue was handledby a group of people in Fig. 1(a), we are interested in passivedynamic object locomotion through alternating rocking andwalking phases. This technique can be suitable for handlingobjects that are too large, bulky to grasp-lift-and-carry orpush, in an energy-efficient manner.

(a)

A

G B

cable #1cable #2

γC

(b)

BG

N

O

W

Fig. 2. (a) Our model with the conic object. (b) The local geometry aroundG, projected onto the vertical plane which is parallel to the vector of gravityg. N denotes the normal force at G and W is the weight of the object.

In this work, our objects of interest are modeled as anoblique circular cone (Fig. 2). The apex of the cone isdenoted A. It is supposed to be the point to be handled bya robot through an end-effector such as the cables seen inFig. 2(a), although a different end-effector is used in ourimplementation for this work. Let B (C) be the point onthe circular rim of the base of the cone that is the closest(farthest) from A. G denotes the point on the rim of the basecurrently in contact with the flat ground surface. γ denotes

the dihedral angle between the base and the ground, whichwill be kept nonzero during the motion.

First, we investigate the kinematics and dynamics of therocking motion of the object, that is, how the motion canbe (1) described under the assumption that the position ofA is fixed and the object rolls without slipping, and (2)generated in a passive dynamic manner by gravity. We thenexamine how to concatenate the rocking motions into a wayof walking the object forward.

IV. KINEMATICS AND DYNAMICS OF OBJECT ROCKING

This section addresses how to describe the rocking motionof the conic object and presents the underlying dynamicsmodel.

A. Kinematics

The kinematics of the rocking motion of the object can bedescribed by investigating the structure of the set of all thefeasible positions of G, the ground contact point (Fig. 2),under the assumption that A is fixed in space by the robotand the base rolls without slipping on the flat ground surface.

By considering the rotational symmetry of the object’sconfigurations about the axis of AA′ where A′ is the foot ofperpendicular of A to the flat ground surface, the set of allthe feasible positions of G can be represented as an annuluson the surface (Fig. 3(a)). G is on the inner (outer) circleof the annulus when it happens to be at B (C). While theobject rolls without slipping, G will move along an one-dimensional subset of the annulus. Thus, a foliation of theannulus (the collection of the distinct feasible paths of G onthe set) can be considered in which the area is decomposedinto such one-dimensional subsets, each of which is referredto as a leaf of the foliation. Unless the object slides on thesurface, G cannot move across the leaves.

The leaves of the foliation can be found by integratingthe vector field of allowable velocities established on theannulus. First, consider a line segment B′C ′ traversing theannulus between the inner and outer circles, and perpendic-ular to these circles (Fig. 3(a)). Second, at each point onthe line segment, denoted p ∈ B′C ′, a unit vector v(p) isassigned such that it represents the direction of the feasiblevelocities when G coincides with p (Fig. 3(b)). The directioncan be found by intersecting the plane of the ground surfaceand the plane of the base of the object. It can be representedusing the vector cross product:

v(p) =bs × gs

||bs × gs||(1)

where bs (gs) denotes the unit normal vector to the base(ground), with respect to the space frame {s}, when Gcoincides with p. gs is a constant vector relative to the spaceframe {s}. bs can be found by multiplying bb (the vector brelative to the body frame {b} fixed to the object, which isa constant vector) and the rotation matrix between {b} and{s}, which is uniquely determined by (1) the orientation ofthe line connecting A and p, which will determine two of thethree unknowns of the rotation matrix, and (2) the tangency

(a) (b) (c) (d)

A

A′ B′

C′

G = p

A′B′ C′

v

A′B′ C′

A′+ + +

Fig. 3. (a) The annulus with the inner (colored blue) and the outer circle (colored green) represents the set of all the feasible positions of G on the flatsurface. A′ is the foot of perpendicular of A to the surface. B′C′ is a line segment normal to the circles. (b) For each p ∈ B′C′, the direction of thefeasible velocities v(p) is computed. (c) The vector field on B′C′ is parallel transported to fill the entire annulus, and the integral curves are obtained.(d) Integral curves describing the local motions about the points on the inner circle.

of the base to the ground surface (the remaining, thirdunknown). Third, by taking into consideration the rotationalsymmetry of the system, the vector field on B′C ′ is paralleltransported along the concentric arcs to fill the annulus, andthe leaves of the foliation are obtained by finding the integralcurves of the vector field over the annulus (Fig. 3(c)). Theexistence of the integral curves is confirmed by consideringthat the vector field on B′C ′ is differentiable (from Eq. 1)and the parallel transport along the concentric arcs is also adifferentiable operation.

In addition to the “counterclockwise” vector field shownin Fig. 3(c), along which the object rolls counterclockwiseabout A′, the clockwise vector field can also be constructedby flipping the vector field. The integral curves of these twovector fields can smoothly be connected at the points on theinner circle. Fig. 3(d) shows these connected integral curves“touching” the inner circle. They now fully depict the motionof the object when G is near the inner circle, on which staticequilibrium is obtained. The motion can happen in either oneof the two directions along the curves.

B. Model of Dynamics

While the object is in motion along the integral curve, asshown in Fig. 3(d), it can be seen that the moment of theforce of gravity mg about the axis AG cannot be balancedunless G = B (the case in which G happens to be onthe inner circle of the annulus) due to the asymmetry ofthe object shape modeled as an oblique cone. This staticimbalance causes the object to dynamically rock about thepoint on the inner circle. Fig. 4 (elaborating Fig. 2(b)) modelsthe situation as a planar rigid rocker with point mass and acurved contour on the plane perpendicular to the flat groundsurface, as seen from the observer at A′ (recall Fig. 3(a)).Provided that O (the center of curvature at B) is above thecenter of gravity, the model predicts that the object willrock sideways about B (the equilibrium position), which isqualitatively correct, like the simple pendulum.

Next, we derive the equation of motion of our reducedplanar rigid rocker with point mass model. See Fig. 4. Theweight of the object is represented as mg sinα, where α

G = B

O

θ a R

mg sinα

y

x

Fig. 4. Planar rigid rocker with point mass. Note that α is the anglebetween the gravity vector and the axis of rotation AG (see Fig. 3(a)).

is the angle between the gravity vector and the axis AGwhen G = B, because only the component of the gravityvector perpendicular to AG accounts for the motion of theobject. θ denotes the angular displacement of the object: it ispositive (negative) if the rocker is displaced to the right (left)of B. Assuming that the radius of curvature of the rocker,R, remains constant, the position of the rocker’s center ofmass (CM) is:

xCM = Rθ − a sin θ and yCM = R− a cos θ

The Lagrangian of this mechanical system is:

L = T − V

=1

2m(x2CM + y2CM

)−m(g sinα)yCM

=1

2m((Rθ − aθ cos θ)2 + (aθ sin θ)2

)−m(g sinα)(R− a cos θ)

The Euler-Lagrange equation of motion is then:(R2 − 2aR cos θ + a2

)θ + (Ra sin θ) θ2

− a(g sinα) sin θ = 0 (2)

One way to see how close the model is to the realityis to compare the model-based and the observed periods of

←−Robot

←−

IMU

←−

Conicobject

α (the angle between the gravity vector and AG) (deg)

Peri

od(s)

Fig. 5. Comparing the theoretical and empirical periods of rockingmotion of the conic object. The theoretical values are obtained by Eq. (2).Empirically, the periods are measured by using an IMU attached to thecone-shaped object as shown above. The robot is simply holding the object,with no controlled motion.

oscillation. This information on the period can also be usefulfor gait generation and control to be discussed later in thepaper. By linearizing Eq. (2) about the equilibrium point, weobtain an expression for the period of small oscillations:

T (α) = 2π

√(R− a)2ag sinα

We then compare this to empirically measured periods ofsmall oscillations of our object for different values of α(Fig. 5). Note that for our choice of conic object, α = γ,where γ is the dihedral angle (Fig. 2), because the verticalbar of the object is normal to the base rim. From Fig. 5, it isevident that the observed cadence is generally greater thanthat expected theoretically. We suspect that the discrepancywas caused mainly by damping induced by rolling friction,which also made it hard to measure the period when α issmall.

V. GAIT GENERATION: ROCK AND WALK

In this section, we examine how elemental rocking stepscan be concatenated to yield the locomotion of the object.

A. Concatenating Steps

It is possible to control the object to move a long distanceby concatenating individual rocking motions, similarly to aLie bracket motion [2]. Fig. 6 shows that the four smoothintegral curves are connected sequentially, which results inthe zigzag path for the ground contact point, G. Each smoothintegral curve will be referred to as a step from this pointon. Fig. 6 thus shows a sequence of four steps. Duringeach step, the object rolls in a passive dynamic mannerfollowing the integral curve; no control efforts are necessary.The object accelerates (decelerates) as G approaches (leaves)the inner circle. At the end of step i, the velocity of the

x

y

step 1

step 2

step 3

step 4

A′1

A′2

A′3

A′4

Fig. 6. Concatenating steps, each of which is shown as the solid arrowrepresenting the integral curve, which is actually the same as the one shownin Fig. 3(d). G will move along the piecewise smooth trajectory, whichdarkens as the steps move forward.

object becomes zero and then A is relocated such that A′i(the foot of perpendicular of Ai, which is the position ofA for step i) moves to A′i+1. The x- and y-coordinatesof A determine the position of A′i+1, in other words, thecenter of the new annulus. The z-coordinate of A determinesthe size of the new annulus, the radii of the inner and theouter circles. Therefore, it is possible to steer the object andadjust the length (or duration) of each step by appropriatelychanging the way A is relocated. For example, the steps ofFig. 6 were generated by (1) keeping the z-coordinates of Aconstant, which results in the series of identical (up to therigid body transformation on the plane) integral curves, and(2) changing the x- and y-coordinates of A such that thelines passing through A′i and A′i+2 are parallel to the y-axisof the reference frame, which makes the object move in thepositive y-direction eventually. Of course, it is also possibleto make turns by aligning A′iA

′i+2 with the desired direction.

The position of A can thus be thought of a “steering wheel”of the object.

B. Robot Control Strategy for Rock-and-Walk Locomotion

The locomotion pattern of the object described in Sec. V-A, which will be called a rock-and-walk gait, can be realizedin a range of ways. One obvious approach is to use theconventional, position-controlled robot arm for once-per-stepadjustments to the position of A. Given a sequence of stepsplanned in advance, each step i determines the position ofA′i (see Fig. 6), and in turn a well-defined collection of thewaypoints for A can be obtained. The position-controlledarm can then be applied to moving A through the waypoints.The transitions between the waypoints need to be performedin a timely manner. For example in Fig. 6, the transitionfrom A′2 to A′3 is supposed to happen at the end of step 2

when the velocity of the object vanishes. If the transitioncan be completed immediately, in principle it is possible toexactly follow the zigzag path shown in Fig. 6. The timewhen the velocity of the object vanishes can be detected bya range of sensing modalities, for example, visual perception.Without the sensor feedback, the information on the objectdynamics (Fig. 5) can be used for the detection of the time,but the errors can be large. In an ideal case with no energydissipation, this operation featuring position control can berepeated continuously.

Some variants of the basic approach can also be consid-ered. For example, multiple robot arms can be coordinated torock and walk the object using cables similarly to Fig. 1(a).To compensate for energy losses, energy shaping [20] controlcan be incorporated as can also be seen in the swing-upcontrollers for the acrobot and cart-pole systems [21], [22].

VI. IMPLEMENTATION AND EXPERIMENTATION

This section describes our hardware and software, andpresents a set of experiments for our rock-and-walk loco-motion.

A. Our Hardware and Software

The object to be used in our experiments is fabricatedby welding a steel rim of radius 0.35m and a steel rodof length 1.5m, along with two support rods on the sides(see Fig. 7). The weight of the object is 1.5kg. Althoughthe object is contrived for demonstration here, it actuallyrepresents a class of objects, for example, cones, to whichour rock-and-walk locomotion technique can be applied. Aninertial measurement unit (IMU, Arduino 9 Axes MotionShield) is fixed to the object and provides orientation andangular velocity data. Universal Robots UR10 six-DOF robotarm is used to manipulate the object. A single rigid-bodyend-effector is fixed to the wrist of the robot arm. The end-effector has no internal mobility, but it can cage the objectusing a hole to accommodate the vertical rod of the object(see [23] for a relevant discussion about caging/immobilizingusing such effectors). The diameter of the hole is 6cm, which

(a) (b)

Fig. 7. (a) Our hardware setting visualized in the RViz environment. Theworld frame and the body-fixed IMU frame are also shown. (b) Our realhardware setting.

WorldX

Y

Z

IMUx

y

z

(a)

(b)

β(r

ad)

β(r

ad)

Time (s)

Time (s)

Fig. 8. Time evolution of the orientation of the object when our controllerwas (a) on and (b) off (here the oscillations attenuate after a few rocks),with the parameters shown in the first row of Table I. β refers to the angulardisplacement of the object about the y-axis of the IMU frame at time t = 0.

is much larger than that of the rod (0.9cm). The object canthus rock freely while being handled by the arm; in otherwords, the object is manipulated in a nonprehensile manner.Our overall hardware setup is depicted in Fig. 7.

We developed MATLAB software that can generate thetrace of steps (the concatenated integral curves) given thegeometry and configuration of the object, and the magnitudeof rocking. The information on the generated steps is thensent to our arm control software organized as a ROS2

package written in Python. It enables the arm to movethrough the sequence of the waypoints that can result in thepre-computed steps. The software can also read data fromthe IMU to implement feedback control.

B. Experiment: One-Armed Rock-and-Walk Locomotion

The object is manipulated to rock and walk using thehardware and software setting. First, pick a point on thevertical bar of the object to be regarded as A. The end-effector is supposed to make contact at A. Also pick a point,denoted X , on the base of the object. The object is supposedto be rocked up to X . Now choose the height of the end-effector, denoted hee, from the ground surface, and hold theobject manually such that A is at the end-effector and Xis in contact with the ground. The object is then releasedand our feedback controller takes control of it. At the nextmoment the angular velocity of the object becomes zero, theend-effector (and thus the position of A) is relocated to thenext, pre-computed position with the same height hee; andthis operation is repeated to continue walking the object.

Fig. 8(a) shows how the orientation of the IMU referenceframe (see Fig. 7(a)) evolves during a rock-and-walk gait inwhich AB = 1.35m, hee = 1.26m, and the length of arcBX (or the magnitude of rocking) 14◦; see the first row inTable I. The robot is stopped after 40 rocking steps due toits limited workspace, around time t = 60s. Fig. 8(b) showsthat the uncontrolled, free oscillation of the object attenuatesquickly. This shows the ability of our controller to sustain therocking motion. Fig. 9 shows some snapshots of the motionof the object; also see the video attachment. The average net

2http://www.ros.org

TABLE IEXPERIMENTAL RESULTS

AB(m) hee(m) BX(◦) steps dpred(m) dactual(m) duration(s) average forward speed(cm/s)1.35 1.26 14 40 0.43 0.40± 0.02 62± 1 0.63

1.35 1.26 14 20 0.22 0.21± 0.01 30± 1 0.701.35 1.20 15 20 0.10 0.12± 0.02 30± 1 0.381.35 1.15 16 20 0.08 0.07± 0.01 29± 1 0.24

1.30 1.26 11 10 0.10 0.15± 0.03 15± 1 1.021.30 1.20 14 20 0.17 0.17± 0.02 30± 1 0.551.30 1.15 16 20 0.21 0.14± 0.02 30± 1 0.45

1.25 1.20 12 10 0.15 0.17± 0.01 15± 1 1.101.25 1.15 14 20 0.18 0.21± 0.03 30± 1 0.68

forward displacement of the object over three trials (denoteddactual in Table I) is 0.40±0.02 meters (1.14 times the radiusof the base) and the time to complete this manipulation is62± 1 seconds on average, with the parameter setting.

(a) t = 0.0s (b) t = 0.8s (c) t = 1.4s

(d) t = 2.1s (e) t = 2.9s (f) t = 3.9s

(g) t = 4.4s (h) t = 5.2s (i) t = 5.9s

Fig. 9. Snapshots showing four rock-and-walk steps with relative timestamps. Also see the video attachment.

On the other hand, the nominal trajectory of G (the contactpoint on the ground) can be obtained numerically using ourMATLAB software, given the geometry of the object and thepositions of A and X . For example, Fig. 10 shows howthe position of G evolves on the ground nominally, giventhe same setting again: AB = 1.35m, hee = 1.26m, andarc BX = 14◦. The figure also shows the correspondingpositions of Ai’s that can derive the motion. Here the net for-ward displacement of the object (denoted dpred in Table I) iscomputed to be 0.43m after 40 steps; the actual displacement

Fig. 10. First four rocking steps (out of 40) and the corresponding positionsof A′i’s obtained numerically, expressed relative to the position of A′1. Here,X is at 14◦ from B, AB = 1.35m and hee = 1.26m.

x (m)

y(m

)A′1A′2A′3A′4

step 1step 2 step 3

step 4

was around 0.40m as presented in the previous paragraph.One reason for the discrepancy between the actual and thepredicted forward displacements is that the object may slidewithin the end-effector during the course of manipulation,and thus the length of AB changes.

The results of our experiments and numerical simulationson several other settings are summarized in Table I. For eachsetting, three trials were conducted. It can be seen from theaverage forward speeds (the rightmost column of the table)that a more upright posture (that is, hee/AB closer to 1)results in faster walking. However, it can also be observedthat as the object gets more upright, the gait becomes lessstable.

VII. CONCLUSION

In this paper we presented a novel nonprehensile manip-ulation technique for passive dynamic object transportation,which we call rock-and-walk locomotion. Possible directionsfor future work include (1) optimal control for maximizingthe speed of walking without losing stability and (2) cable-driven implementation (as in Fig. 1(a)), which can extend therange of objects and robot platforms to which this techniquecan be applied.

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