Chapter 3. Kinematic analysis

Table of Contents Kinematics Velocity Kinematics

3.1 Kinematics Kinematics is describing the geometrical motion of the robot without considering the force/torque or inertia. Forward kinematics: Calculating the position/orientation of the end-effector based upon the joint angles. Inverse kinematics: Calculating the joint angles for the desired end-effector position/orientation.

Analysis of manipulators Manipulator: Joint i + Link i + Joint i+1 + Link i+1 Structure. The links are connected in serial generally, which have linear or angular motions by the actuator. Analysis of manipulator motion: The relationship of the moving frame at the end of the link with respect to the reference frame.

General representations Robot has n+1 links: from 0 to n from the base of the robot to the end-effector. Joint number: 1  n , Joint i is located at the beginning of link i . ith Joint: Revolute joint: Prismatic joint: Frames: at the end of each link.

Example 3.1 [Fig. 3-1] 3 DOF Cylindrical manipulator

[Table 3-1] Link parameter of 3-DOF manipulator Link Parameter Table Following the D-H constraints, the table is obtained as follows: -90 Link 1 2 3 [Table 3-1] Link parameter of 3-DOF manipulator

Homogeneous Transformation Matrix (1. 1) (1. 2) (1. 3)

Composition (1. 4)

Example 3.2 [Fig. 3-2] 6 DOF PUMA 560 Manipulator

D-H Frames [Fig. 3-3] Frames for PUMA 560 Manipulator

[Table 3-2] Link parameter of 6 DOF PUMA 560 manipulator Link Parameter Table Link 1 2 -90 3 4 5 90 6 [Table 3-2] Link parameter of 6 DOF PUMA 560 manipulator

Homogeneous Transformation Matrix (1. 5) (1. 6) (1. 7)

Homogeneous Transformation Matrix (1. 8) (1. 9) (1.10)

Composition (1. 11) (1. 12)

Composition (1.13)

Composition (1.14)

Composition (1.15) (1.16) (1.17) (1.18) 2019-07-18

Composition Finally for the six links, (1.19) (1.20)

Composition (1.21) (1.22) (1.23)

3.2 Velocity kinematics When a matrix, S, satisfies the following conditions: where the matrix S is skew symmetric. (1.24) (1.25)

Skew symmetric (1.26) (1.27) (1.28) (1.29) where,

Properties of skew symmetric matrix Vectors and are in ; and are scalars, (1.30) For a vector, (1.31)

Cross Product When and vectors are in (1.32) is satisfied and R is Orthogonal. (1.32)

Cross Product When and vectors are in (1.33) where R is Orthogonal.

Cross Product (1.34)

Proof Using the chain rule, derivative of w.r.t (1.35) (1.36) (1.37)

Skew symmetric (1.38) Matrix is skew symmetric. Multiplying at both side of , we have (1.39)

Angular velocity (1.40)

Derivative of Velocity The vector can be represented w.r.t the frame as, Then becomes, (1.41) (1.42)

3.2.1 Velocity propagation [Fig. 3-4] Vector representation In Fig. 3-4 , how the vector will be changed?

Derivative of Motion Vector can be differentiated w.r.t time with fixed (1.43) where is a vector.

Proof Show that [Fig. 3-5] Rotation about an arbitrary axis k .

Composition results (1.44) can be obtained by differentiating each element as follows:

Motion Vector Differentiation the first element is (1.45) where is scalar. (1.46)

Obtaining Let’s obtain (1.47) (1.48)

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Derivation (1.49) (1.50)

Derivation (1.51) Therefore . (1.52)

Derivation Since , (1.53) (1.54) (1.55)

3.2.2 Propagation of angular velocity Vector can be represented by as  (1.56)

Angular velocity (1.57) (1.58)

Angular velocity (1.59) (1.60)

Addition of Angular Velocity (1.61) (1.62)

Addition of Angular Velocity (1.63) To generalize this equation, is adopted. (1.64)

3.2.3 Jacobian Robot generates motion at each joint, which results motion at the end-effector w.r.t frame {0}. The motions are related by J(Jacobian) as (1.65)

Obtaining Jacobian . Using in can be obtained w.r.t frame {0} That is, . (1.66)

Jacobian  (1.67) (1.68) (1.69)

Obtaining Jacobian where is the rotation axis of th joint. When the th joint is linear, The rotational motion at the end-effector can be represented as the summation of the rotation motions w.r.t {0} frame. (1.70)

Jacobian The rotation axis is , that is, Therefore (1.71) gives easily.

Jacobian Finally (1.72) Generally , is named as Jacobian.

3.2.4 Acceleration Propagation [Fig. 3-6] Vector representation

Acceleration is time-varying, then (1.74) (1.75) Taking the derivative at both sides,

Acceleration (1.76) First term: Transverse acceleration Second term: Centripetal acceleration Third Term: Coriolis acceleration Fourth Term: Linear acceleration

Acceleration Finally, it can be represented as (1.77) It is very complex. Generally, is used to represent the accelerations by taking derivative at both sides.

[Fig. 3-7] 2 Link manipulator Example 3.3 Jacobian matrix [Fig. 3-7] 2 Link manipulator

[Table 3-3] Link parameter of 2 Link manipulator Link Parameter Table 2 1 Link [Table 3-3] Link parameter of 2 Link manipulator

Homogeneous matrix (1.78) (1.79)

Composition (1.80) (1.81)

Obtaining Jv (1.82) (1.83)

Obtaining Jω (1.84) (1.85)

Final Jacobian matrix (1.86)

Example 3.4 [Fig. 3-8] SCARA robot

[Table 3-4] Link parameter of SCARA robot Link Parameter Table Link 1 2 180 3 4 [Table 3-4] Link parameter of SCARA robot

Homogeneous matrix (1.87) (1.88) (1.89)

Composition (1.90) (1.91)

Each element (1.92) (1.93) (1.94) (1.95)

Obtaining Jv (1.96) (1.97) (1.98) (1.99) (1.100) (1.101)

Obtaining Jv (1.102)

Obtaing Jω (Method 1) Method 1. Using (1.103)

Obtaining Jω (Method 2) Using Method 2. (1.104)

Obtaining Jω (Method 2) (1.105) (1.106) (1.107)

Cont. can be obtained (1.108)

Cont. (1.109) (1.110)

Cont. (1.111) Plugging into

Cont. (1.112) Finally, the Jacobian matrix is (1.113)

Final Jacobian matrix (1.114)

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