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control-notes - CS 685 notes, J. Koeck sa 1 Trajectory...

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CS 685 notes, J. Koˇseck´a 1 Trajectory Generation The material for these notes has been adopted from: John J. Craig: Robotics: Mechanics and Control . This example assumes that we have a starting position and goal pose of the end effector and we are asked to move the joint angles to move the end effector from one pose to another. Here we describe a strategy how to do so by designing a trajectory in joint space from one end point to another. Assuming that we know the inverse kinematics of the system, we can compute the desired joint angle for goal position of the end effector. This example shows how to design a trajectory of a single joint θ ( t ) as function of time. Suppose that we have following constrains of out trajectory: we have desired position at the beginning and end of the trajectory and we the velocity at the beging and end has to be zero. Hence our desired trajectory has to satisfy the following constraints: θ ( 0 ) = θ 0 ; θ ( t f ) = θ d ˙ θ ( 0 ) = 0 ˙ θ ( t f ) = θ d (1) Cubic polynomials In order to satisfy the above constraints, our trajectory has to be at least polynomial of the 3 r d order, which has four coefficients, and hence can satisfy the above 4 constraints. This can be achieved by third order cubic polynomial which has the following form θ ( t ) = a 0 + a 1 t + a 2 t 2 + a 3 t 3 . Given the above form the joint velocity and acceleration will have the following forms ˙ θ = a 1 + 2 a 2 t + 3 a 3 t 2 (2) ¨ θ = 2 a 2 + 6 a 2 (3) 1
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Using the above equations and instantiating the constraints we can solve for the coefficients of the cubic polynomial and obtain a 0 = θ 0 (4) a 1 = 0 (5) a 2 = 3 t 2 f ( θ f - θ 0 ) (6) a 3 = 2 t 3 f ( θ f - θ 0 ) (7) Now given a particular instance of the problem, we can substitute to the above equations the desired parameters θ 0 , θ f , t f and obtain different trajectories. Linear functions with parabolic blends If we were to simply just connect the desired position with a linear function, it would cause the velocity to be discon- tinuous at the beginning and end of the motion. Also note that of the shape of the part in the joint space is linear, that does not mean that the shape of the path in the end effector space is linear. Hence what can be done is to take a linear path in the end effector space and interpolate it linearly. We would like to do it in a way that the velocities at the would not be discontinuous at the places where the pieces meet. One way to achieve this is to add a parabolic blend region, such
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This note was uploaded on 04/07/2010 for the course CS 685 taught by Professor Luke,s during the Fall '08 term at George Mason.

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control-notes - CS 685 notes, J. Koeck sa 1 Trajectory...

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