20. Position Jacobian : get from Homogeneous Transformation Matrix
Orientation Jacobian : get a last row of Rotation matrix
Inverse Kinematics
6 DOF Robot
22. 𝑥1, 𝑦1
𝑥2, 𝑦2
Inverse Kinematics
Generalized IK using Jacobian
Piecewise Linearization
- Not only the velocity of joint angles and end effector, but
also the position of them can be estimated using Jacobian
- Jacobian is effective under the condition that angular and
positional motions are small Piecewise linearization
24. Importance of Jacobian
Kinematics (mapping of changes from joint to task space)
• Inverse kinematics control
• Resolve redundancy problems
• Express contact constraints
Statics (and later also dynamics)
• Principle of virtual work
Variations in work must cancel for all virtual displacement
Internal forces of ideal joint don’t contribute
25. Singularities
A singularity is a joint-space configuration such that is column-
rank deficient
• the Jacobian becomes badly conditioned
• small desired velocities produce high joint velocities
Use a damped version of the Moore-Penrose pseudo inverse
Minimize norm of joint angular velocity
26. Redundancy
A kinematic structure is redundant if the dimension of the task-space is
smaller than the dimension of the joint-space
E.g. the human arm has 7DoF (three in the shoulder, one in the
elbow, and three in the wrist)
Many solutions per problem.
Which one to pick ?
27. Min Norm Null space : internal motion,
not effective to the motion of
end effector
Arbitrary
x
Span
Min Norm Null space
Redundancy
r : rank of JE
35. Inverse Dynamics Control
• Model based Torque estimation
• In case of no modeling errors,
• the desired dynamics can be perfectly prescribed
Model
Can achieve great performance…
But requires accurate modeling
𝜏
36. Inverse Dynamics Control
• In real world, modeling errors are always present
• Small error due to modeling error can be compensated
37. Path and Trajectory Planning
Trajectory considers not only the path from A to B
but also the time, velocity, etc
39. Path and Trajectory Planning
After generation of trajectory of end effector at work space(Cartesian space),
the trajectories of joints can be calculated using inverse kinematics
3rd order polynomial is sufficient if position and velocity are considered
5th order polynomial is needed if acceleration are included
40. Path and Trajectory Planning
- 5th or polynomial 6 unknowns 6 equations are needed
- Can get a unique solution for a given 6 initial and terminal conditions
