Robotics and Dynamics

C. Sivakumar Assistant Professor Department of Mechanical Engineering BSA Crescent Institute of Science and Technology

Robot Kinematics and 1 Dynamics_Sivakumar_C kinematics • KINEMATICS – the analytical study of the geometry of motion of a mechanism: – with respect to a fixed reference co-ordinate system, – without regard to the forces or moments that cause the motion.

• In order to control and programme a robot we must have knowledge of both its spatial arrangement and a means of reference to the environment.

Robot Kinematics and 2 Dynamics_Sivakumar_C Open Chain Kinematics • Mechanics of a manipulator can be represented as a of rigid bodies (links) connected by revolute or prismatic joints.

• One end of the chain is constrained to a base, while an end effector is mounted to the other end of the chain.

• The resulting motion is obtained by composition of the elementary motions of each link with respect to the previous one

Robot Kinematics and 3 Dynamics_Sivakumar_C Robot kinematics

• Joint labeling: started from 1 and moving towards end effector, base being joint 1

Robot Kinematics and 4 Dynamics_Sivakumar_C Two Basic Joints

Robot Kinematics and 5 Dynamics_Sivakumar_C Position representation • Kinematics of RR robot is difficult compared to LL robot • Analyzing in 2-D

Robot Kinematics and 6 Dynamics_Sivakumar_C • Position of end of the arm can be represented using: • Joint space method: using joint angles

• World space : using cartesian coordinate system.

Robot Kinematics and 7 Dynamics_Sivakumar_C • Transformation from one representation to other is necessary for many application. Type of transformation: • Forward transformation or forward kinematics – going from joint space to world space • Reverse transformation or – going from world space to joint space.

Robot Kinematics and 8 Dynamics_Sivakumar_C • Direct (also forward) kinematics – Given are joint relations (rotations, translations) for the robot arm. Task: What is the orientation and position of the end effector? • Inverse kinematics – Given is desired end effector position and orientation. Task: What are the joint rotations and orientations to achieve this?

Robot Kinematics and 9 Dynamics_Sivakumar_C Robot Kinematics and 10 Dynamics_Sivakumar_C Robot Kinematics and 11 Dynamics_Sivakumar_C Robot Kinematics and 12 Dynamics_Sivakumar_C Reverse transformation of 2 DOF arm

Robot Kinematics and 13 Dynamics_Sivakumar_C Robot Kinematics and 14 Dynamics_Sivakumar_C Figure 4-4

Robot Kinematics and 15 Dynamics_Sivakumar_C Robot Kinematics and 16 Dynamics_Sivakumar_C Robot Kinematics and 17 Dynamics_Sivakumar_C 3 DOF arm in two dimension

Robot Kinematics and 18 Dynamics_Sivakumar_C 3 DOF arm in two dimension

Robot Kinematics and 19 Dynamics_Sivakumar_C 4 DOF manipulator in three dimensions

Robot Kinematics and 20 Dynamics_Sivakumar_C Robot Kinematics and 21 Dynamics_Sivakumar_C Robot Kinematics and 22 Dynamics_Sivakumar_C Robot Dynamics

• Accurate control of manipulator depends on precise control of joints • Control of joints depends on forces and intertias acting on them

Robot Kinematics and 23 Dynamics_Sivakumar_C a. Static analysis

Robot Kinematics and 24 Dynamics_Sivakumar_C Balancing the forces to know the torque

Robot Kinematics and 25 Dynamics_Sivakumar_C Compensating for gravity

Robot Kinematics and 26 Dynamics_Sivakumar_C Robot arm dynamics

Robot Kinematics and 27 Dynamics_Sivakumar_C Torque requirement

Robot Kinematics and 28 Dynamics_Sivakumar_C Kinematic • Forward (direct) Kinematics • Given: The values of the joint variables. • Required: The position and the orientation of the end effector.

• Inverse Kinematics • Given : The position and the orientation of the end effector. • Required : The values of the joint variables.

Robot Kinematics and 29 Dynamics_Sivakumar_C Why DH notation

• Find the homogeneous transformation H relating the tool frame to the fixed base frame

Robot Kinematics and 30 Dynamics_Sivakumar_C Why DH notation

• A very simple way of modeling robot links and joints that can be used for any kind of robot configuration.

• This technique has became the standard way of representing and modeling their motions.

Robot Kinematics and 31 Dynamics_Sivakumar_C DH Techniques

1. Assign a reference frame to each joint (x-axis and z-axis). The D-H representation does not use the y-axis at all.

2. Each homogeneous transformation Ai is represented as a product of four basic transformations

Robot Kinematics and 32 Dynamics_Sivakumar_C DH Techniques

• Matrix Ai representing the four movements is found by: four movements 1. Rotation of  about current Z axis 2. Translation of d along current Z axis 3. Translation of a along current X axis 4. Rotation of  about current X axis A Rot Trans Trans Rot i z,,,,i z d i x a i x i

Robot Kinematics and 33 Dynamics_Sivakumar_C 1 0 0  C  S 0 R  Rot(x,)  0 C  S  R  Rot(z,)  S C 0 x,   z,   0 S C   0 0 1

Ci  Si 0 01 0 0 0 1 0 0 ai 1 0 0 0 S C 0 00 1 0 0 0 1 0 0 0 C  S 0  i i    i i  Ai   0 0 1 00 0 1 di 0 0 1 0 0 Si Ci 0       0 0 0 10 0 0 1 0 0 0 1 0 0 0 1

ci -c  i s  i s  i s  i a i c  i s c  c  -s  c  a s  i i i i i i i Ai  0 si c i d i  0 0 0 1 Robot Kinematics and 34 Dynamics_Sivakumar_C DH Techniques

• The link and joint parameters :

• Link length ai : the offset distance between the Zi-1 and Zi axes along the Xi axis.

• Link offset di the distance from the origin of frame i−1 to the Xi axis along the Zi-1 axis.

Robot Kinematics and 35 Dynamics_Sivakumar_C DH Techniques

•Link twist αi :the angle from the Zi-1 axis to the Zi axis about the Xi axis. The positive sense for α is determined from zi-1 and zi by the right-hand rule.

•Joint angle θi the angle between the Xi-1 and Xi axes about the Z axis. i-1 Robot Kinematics and 36 Dynamics_Sivakumar_C DH Techniques

• The four parameters: ai: link length, αi: Link twist , di : Link offset and

θi : joint angle.

• The matrix Ai is a function of only a single variable qi , it turns out that three of the above four quantities are constant for a given link, while the fourth parameter is the joint variable.

Robot Kinematics and 37 Dynamics_Sivakumar_C DH Techniques

th • With the i joint, a joint variable is qi associated where

All joints are represented by the z-axis. • If the joint is revolute, the z-axis is in the direction of rotation as followed by the right hand rule. • If the joint is prismatic, the z-axis for the joint is along the direction of the liner movement.

Robot Kinematics and 38 Dynamics_Sivakumar_C DH Techniques

3. Combine all transformations, from the first joint (base) to the next until we get to the last joint, to get the robot’s total transformation matrix.

0 TAAAnn 12......

0 4. From T n , the position and orientation of the tool frame are calculated.

Robot Kinematics and 39 Dynamics_Sivakumar_C DH Techniques

Robot Kinematics and 40 Dynamics_Sivakumar_C DH Techniques

Robot Kinematics and 41 Dynamics_Sivakumar_C DH Techniques

Robot Kinematics and 42 Dynamics_Sivakumar_C DH Techniques

ci -c  i s  i s  i s  i a i c  i s c  c  -s  c  a s  i i i i i i i Ai  0 si c i d i  0 0 0 1

Robot Kinematics and 43 Dynamics_Sivakumar_C