1. Review: Differential Kinematics
Find the relationship between the joint velocities and the end-effector linear and angular velocities.
Linear velocity
Angular velocity
for a revolute joint
for a prismatic joint

2. Review: Differential Kinematics
Approach 1

3. Review: Differential Kinematics
Approach 2
Prismatic joint
Revolute joint

4. Review: Differential Kinematics
Approach 3
The contribution of single joint i to the end-effector linear velocity
The contribution of single joint i to the end-effector angular velocity

5. Review: Differential Kinematics
Approach 3

6. Kinematic Singularities
The Jacobian is, in general, a function of the configuration q; those configurations at which J is rank-deficient are termed Kinematic singularities.

7. Reasons to Find Singularities
Singularities represent configurations at which mobility of the structure is reduced
Infinite solutions to the inverse kinematics problem may exist
In the neighborhood of a singularity, small velocities in the operational space may cause large velocities in the joint space
此处取J＝diag(1,0.0001)，即便要求v=[1,1]，关节速度也会非常大

8. Problems near Singular Positions
The robot is physically limited from unusually high joint velocities by motor power constraints, etc. So the robot will be unable to track this joint velocity trajectory exactly, resulting in some perturbation to the commanded cartesian velocity trajectory
The high accelerations that come from approaching too close to a singularity have caused the destruction of many robot gears and shafts over the years.

9. Classification of Singularities
Boundary singularities that occur when the manipulator is either outstretched or retracted.
Not true drawback
Internal singularities that occur inside the reachable workspace
Can cause serious problems

10. Example 3.2: Two-link Planar Arm
Consider only planar components of linear velocity
Consider determinant of J
Conditions for singularity

11. Example 3.2: Two-link Planar Arm
Conditions for sigularity
Jacobian when theta2=0

12. Singularity Decoupling
Computation of internal singularity via the Jacobian determinant
Decoupling of singularity computation in the case of spherical wrist
Wrist singularity
Arm singularity

13. Singularity Decoupling
Wrist Singularity
Z3, z4 and z5 are linearly dependent
Cannot rotate about the axis
orthogonal to z4 and z3

14. Singularity Decoupling
Elbow Singularity
Similar to two-link planar arm
The elbow is outstretched or retracted

15. Singularity Decoupling
Arm Singularity
The whole z0 axis describes a continuum of singular configurations

16. Singularity Decoupling
Arm Singularity
A rotation of theta1 does not cause any translation of the wrist position
The first column of JP1=0
Infinite solution
Cannot move along the z1 direction
The last two columns of JP1 are orthogonal to z1
Well identified in operational space;
Can be suitably avoided in the path planning stage
前面的两个例子奇异条件都是关节变量；而此例子奇异条件是用操作变量。

17. Differential Kinematics Inversion
Inverse kinematics problem:
there is no general purpose technique
Multiple solutions may exist
Infinite solutions may exist
There might be no admissible solutions
Numerical solution technique
in general do not allow computation of all admissible solutions
利用微分运动学可以解逆运动学问题

18. Differential Kinematics Inversion
Suppose that a motion trajectory is assigned to the end effector in terms of v and the initial conditions on position and orientations
The aim is to determine a feasible joint trajectory (q(t), q’(t)) that reproduces the given trajectory
Should inverse kinematics problems be solved?

19. Differential Kinematics Inversion
Solution procedure:
If J is not square? (redundant)
If J is singular?
If J is near singularity?
0时刻，q已知，可求出q(0)，假定在【0，delta T】关节做匀速运动，可得到q(delta T),如此迭代即可。

20. Analytical Jacobian
The geometric Jacobian is computed by following a geometric technique
Question: if the end effector position and orientation are specified in terms of minimal representation, is it possible to compute Jacobian via differentiation of the direct kinematics function?

21. Analytical Jacobian
Analytical technique

22. Analytical Jacobian Analytical Jacobian For the Euler angles ZYZ
证明思路：从R(T)倒数和S的关系入手

23. Analytical Jacobian
From a physical viewpoint, the meaning of ώ is more intuitive than that of φ’
On the other hand, while the integral of φ’ over time gives φ, the integral of ώ does not admit a clear physical interpretation

24. Example 3.3

26. Statics
Determine the relationship between the generalized forces applied to the end-effector and the generalized forces applied to the joints - forces for prismatic joints, torques for revolute joints - with the manipulator at an equilibrium configuration.
Gamma为施加在环境上的广义力，tao为产生此效果关节需施加的广义力;例子在黑板上演示（关节角都为0的情况）

27. fy Y0 R y0 fx x2 y2 v a2 Y1 q2 X1 a1 v v q1 X0 v x0

28. Statics
Let τ denote the (n×1) vector of joint torques and γ(r ×1) vector of end effector forces (exerted on the environment) where r is the dimension of the operational space of interest

29. fy Y0 R y0 fx x2 y2 v a2 Y1 q2 X1 a1 v v q1 X0 v x0

30. Manipulability Ellipsoids
Velocity manipulability ellipsoid
Capability of a manipulator to arbitrarily change the end effector position and orientation

31. Manipulability Ellipsoids
Velocity manipulability ellipsoid
Manipulability measure: distance of the manipulator from singular configurations
Example 3.6

32. Manipulability Ellipsoids
Force manipulability ellipsoid

33. Manipulability Ellipsoids
Manipulability ellipsoid can be used to analyze compatibility of a structure to execute a task assigned along a direction
Actuation task of velocity (force)
Control task of velocity (force)

34. Manipulability Ellipsoids
Control task of velocity (force)
Fine control of the vertical force
Fine control of the horizontal velocity

35. Manipulability Ellipsoids
Actuation task of velocity (force)
Actuate a large vertical force (to sustain the weight)
Actuate a large horizontal velocity