INTRODUCTION TO DYNAMICS ANALYSIS OF ROBOTS (Part 6)

This lecture continues the discussion on the analysis of the instantaneous motion of a rigid body, i.e. the velocities and accelerations associated with a rigid body as it moves from one configuration to another. After this lecture, the student should be able to: Analyze problems of evolution of rigid body configuration with time Introduction to Dynamics Analysis of Robots (6)

Application Issues In many industry applications (e.g. welding and assembly), the desired path of the gripper is known. In these applications, it is common to divide the path into many points that the gripper must successively pass through. These points are then used to get the required joint angles based on inverse kinematics solutions. This raises another issue. How fast the gripper traverses these points in succession will depend on the rate of change of the joint variables, i.e. the linear velocity of the gripper will depend on the angular velocities of the joints. This issue is important because for a robot to maneuver in real time, the joint angles and velocities have to be determined quickly for the robot control. The accuracy of the results will ultimately affect the accuracy and repeatability of the robot.

Application Issues Point ‘A’Point ‘B’ Robot path Inverse kinematics can determine the angles  1A,  2A &  3A at point ‘A’. Inverse kinematics can also determine the angles  1B,  2B &  3B at point ‘B’. How fast the arm travel from point ‘A’ to B’ will depend on the rate the corresponding angles  1A,  2A &  3A change to  1B,  2B &  3B.

Trajectory Planning Consider the rotation of the following link: The link starts from rest, i.e. at t 0 =0 The link must stop at t f, i.e. The total time taken for the rotation is At the time t h where the rotation should have covered The mid point!

Trajectory Planning One way to accomplish the rotation is as follow: t t From t 0 =0 to t 1 : constant acceleration Velocity will change from to From t 1 to t 2 : zero acceleration Velocity held constant at From t 2 to t f constant deceleration at Velocity will change from to 0 t

Trajectory Planning (constant) Note: =0 (initially at rest) For t 0  t  t 1 : At t=t 1 : For t 1  t  t 2 : (constant)  will change from  1 at t=t 1 to  2 at t=t 2 where t t t

Trajectory Planning t t t T

t t t T Rearranging: Solving it to get:

Trajectory Planning t t t T Obviously T 1  t h : This is the time required for acceleration and deceleration

Trajectory Planning t t t T To avoid any imaginary solutions: This is the constraint on the value of the constant acceleration

Trajectory Planning t t t T Given  0 and  f and total time “T”, the plan works as follow: Select a value for acceleration Find the “blend time” T 1 using: where Implement the trajectory plan according to the diagrams on the left

Trajectory Planning For t 0  t  t 1 : At t=T 1 : For t 1  t  t 2 : (constant) t t t At t=T 2 : T-2T 1 (constant)

Trajectory Planning For t 1  t  t f : At t=t f =T: t t t T-2T 1 (constant)

Point ‘A’Point ‘B’ Robot path Putting it Altogether We want the gripper to move from point “A” to “B”. Given these points w.r.t. base frame, we can apply the inverse kinematics to derive the joint angles  0 s at point “A” and angles  f s at point “B”. We then have to decide on the time “T” to accomplish the motion. With these parameters, we can apply the trajectory planning discussed.

Other Trajectory Planning Approaches All trajectory planning approaches basically involve generating a path w.r.t. between the given starting point (represented by  0 ) and the ending point (represented by  f ). t T There are many possibilities! The constraints are: One common approach is to use a cubic polynomial to represent the path

Cubic Polynomial The cubic polynomial has the form

Cubic Polynomial

Summarizing:

This lecture continues the discussion on the analysis of the instantaneous motion of a rigid body, i.e. the velocities and accelerations associated with a rigid body as it moves from one configuration to another. The following were covered: Evolution of rigid body configuration with time Summary