robotics: cartesian trajectory planning
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Cartesian Trajectory Planning
• Trajectories can include via points– Pass close to but not necessarily pass through
– (knot points in b-splines: graphics)
• E.g. straight line paths connected by via point– Both position and orientation have to be interpolated
p0
p(t1
p2
p1
p(t1
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Positional Translation
• Shape of the interpolated region ?– RH Taylor 1979– Start at p0, arrive at p1 in time t1 under
constant velocity– End at p2, from at p1 in time t2 under
constant velocity– At time before arrival at p1 begin the
curved transition thus p(t1- ) and p(t1+ ) are the two transition points
– The curved segment is a parabola with constant acceleration
p0
p(t1
p2
p1
p(t1
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• The initial conditions p0, p1, p2, t1and t2 are all specified.
• We now consider the position and velocity constraints of the transition points
p
2
21
1
11
12221
11
01111
11
a (t)pon acceleraticonstant
p
)(p and p
)(p
:sConstraintVelocity
ppp where)(
ppp where)(
:sConstraintPosition
t
tt
t
pt
ptp
pt
ptp
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• Integrating the acceleration equation we get
• Rearrange and express in terms of the position function p(t)
• Solve for ap at the second transition point t=t1+
• Substitute into the equation for p(t)
21111
2
0
)(2
))(()()(
2Vs :onacceleraticonstant under motion using
tta
tttptptp
att
p
211
1
11 )(
2)(
tt
att
t
pptp p
1
1
2
2
2
1
t
p
t
pap
21
2
221
1
11 )(
44)(
tt
t
ptt
t
pptp
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• The solution to the trajectory reduces to:
• Note the solution does not pass through point p1
211
112
12
2
21
1
21
1
11
11
1 0
44
)(
tttt
ttt
tt
ttt
p
pt
ttp
ttt
pp
pt
ttp
tp
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Rotational Transition• The rotational transition is found by finding the equivalent
axis of rotation k
• R0 is the start orientation, R1 the orientation at the via point and R2 the orientation at the goal.
101110 11 RRRRRR T
kk
122
011
212220
to s transform and
to s transform axis-angle where
2
1
22
RRR
RRR
RRRRRR
k
k
Tkk
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• The rotation about k1 and k2 can be made a linear function of time
• Rotation along the straight lines
• The rotations can be derived in a similar way to the positions
21122
11
111
11
,
0 ,
2
1
ttttt
ttRRtR
ttt
ttRRtR
k
k
112
2
21
11
21
1 ,4
)(
4
)(11
tttt
ttR
t
ttRRtR kk
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Velocity and Acceleration• We need to describe the velocities and
accelerations of tools or of grasped objects
• Position of a link rotating about and origin
• Joint angle velocity
• Same as swinging ball velocity perpendicular to position vector
Oo
x1y1
a1
x0
1
r
v
1
111
1101
0
s
caxad
1111
11101
0 yac
sad
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• More general form based on rotation matrices
• Y is found by rotating x by /2
10
1010
1111
11111
0
11
1111
0
11
110
010
11
110
010
2
yd
Rcs
scRR
cs
scRR
xaRdxaRd
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• Consider the more general case where the link length is not fixed
010
11
1100
11
11
1010
00
11
1010
00
2
)0(
dpRp
p
pRdp
pRdp
O1
O0
p0
p1
1
d01
y0
x0
x1y1
PR(+/2)p1
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3D-Motion
• Derivatives of Rotational Matrices
• Euler angle rates– Representational singularities: Some valid
velocities cannot be represented by Euler angles
• Quaternion Rates– Convert from to q and integrate to get q
IqS
q
dt
d T
0
0
)(2
1
q
q
q
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Manipulator Jacobian• Matrix of differentials
• Describe the motion of the tool in terms of changes in the joints
• Jacobian calculated by differentiating the Forward Kinematic transform
CartesianVelocities
JointVelocities
ddx J
dxd 1J
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Inverse Kinematic Velocities and Accelerations
• Given a tool speed…. Find angle rates • Inverse Jacobian Method
– Assumes that the Jacobian is non-singular(has an inverse at all points)
– Not true at Singularities
– Very Computationally expensive
• Block Matrix Method– Split the Jacobian up into components exploiting the
geometries of the robot arm
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Joint Force and Torque• Gravity acts at the centre of mass• Force/torque equations for link 2
• Force /torque equations for link 12
1
a1
a2
O2
O1
O0
x1
x0
x2
y0
m1g
m2g
z0
z1
2link on 1link by exerted troqueand force2
0
0
1212
12212
122
n
ngd
g
f
m
fm
1link on 2link by
exerted troqueand force2
0
0
2121
21012101101
21011
n
nndgd
g
f
fm
ffm
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• Rearranging the equations
• Joint 1 gravity compensation torque compensates for its own weight plus the torque due to link 2
• Forces and torques are generated at the end effector– Again the Jacobian is used to compute the transmitted
forces and torques
gd
dgd
n
g
212
01101
01
2101
22
)(
mm
mmf
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Dynamics
• Newton-Euler equations
IIrp
Irp
I
lnmf
nf
lm
m
Equation sEuler'Equation Newtons
TorqueNet ForceNet
MomentumAngular MomentumLinear
InertiaMass
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• Calculating the dynamic Joint torques – For the planar manipulator the Newton Euler
equations can be derived – Equations used to determine the driving forces
and torques
• Forward Dynamics– Joint Force and Torque Joint Motion
• Inverse Dynamics– Joint Motion Joint Forces and Torques
• General Manipulator dynamics– Recursive application of Newton-Euler
dynamics
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Position Control• Proportional Derivative(PD) control
• Damp the energy out of the motion to stop at end point of path
• K is the proportional part, B is the damping part
• Motion can be • under damped: oscillations• over-damped: sluggish response• or critically damped: best response without
ossilations
dBKdf
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Trajectory following
• Proportional Velocity(PV) control– Remove damping except where there are
deviations from the path
• For manipulator each joint can have an independent PV controller
)()( rr ddBddKf
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Other Schemes
• Computed Torque Control– Feedback at one joint affects the others– Take these effects into accout
• Resolved Acceleration Control– Use quaternions to allow the use of cartesian
controller
• Resolved Acceleration Force Control– Combine the above with the transmitted forces
as detailed by the J matrix
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Manipulator Robotics
• Haptics and Interaction– Force Control and Compliance
• Non Rigid manipulators– Space and surgery
• Task Based and Shared Control– Tele-manipulation