Robotics kinematics and Dynamics C. Sivakumar Assistant Professor Department of Mechanical Engineering BSA Crescent Institute of Science and Technology Robot Kinematics and 1 Dynamics_Sivakumar_C Robot 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 kinematic chain 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 inverse kinematics – 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 robots 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 Ci Si 0 01 0 0 0 1 0 0 ai 1 0 0 0 S C 0 00 1 0 0 0 1 0 0 0 C S 0 i i i i Ai 0 0 1 00 0 1 di 0 0 1 0 0 Si Ci 0 0 0 0 10 0 0 1 0 0 0 1 0 0 0 1 ci -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 si 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 ci -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 si c i d i 0 0 0 1 Robot Kinematics and 43 Dynamics_Sivakumar_C.