Robot Kinematics

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Robot kinematics is the study of the motion (kinematics) of robots. In a kinematic analysis the position, velocity and acceleration of all the links are calculated without considering the forces that cause this motion. The relationship between motion, and the associated forces and torques is studied in robot dynamics. One of the most active areas within robot kinematics is the screw theory.

Robot Kinematics Model

Robot kinematics deals with aspects of redundancy, collision avoidance and singularity avoidance. While dealing with the kinematics used in the robots we deal each parts of the robot by assigning a frame of reference to it and hence a robot with many parts may have many individual frames assigned to each movable parts. For simplicity we deal with the single manipulator arm of the robot. Each frames are named systematically with numbers, for example the immovable base part of the manipulator is numbered 0, and the first link joined to the base is numbered 1, and the next link 2 and similarly till n for the last nth link.Robot kinematics are mainly of the following two types: forward kinematics and inverse kinematics. Forward kinematics is also known as direct kinematics. In forward kinematics, the length of each link and the angle of each joint is given and we have to calculate the position of any point in the work volume of the robot. In inverse kinematics, the length of each link and position of the point in work volume is given and we have to calculate the angle of each joint.

Robot kinematics can be divided in serial manipulator kinematics, parallel manipulator kinematics, mobile robot kinematics and humanoid kinematics.

Forward position kinematics

The forward position kinematics (FPK) solves the following problem: “Given the joint positions, what is the corresponding end effector’s pose?”

Serial chains:

The solution is always unique: one given joint position vector always corresponds to only one single end effector pose. The FK problem is not difficult to solve, even for a completely arbitrary kinematic structure.

Methods for a forward kinematic analysis:

  • using straightforward geometry
  • using transformation matrices

Parallel chains (Stewart Gough Manipulators):

The solution is not unique: one set of joint coordinates has more different endeffector poses. In case of a Stewart Platform there are 40 poses possible which can be real for some design examples. Computation is intensive but solved in closed form with the help of algebraic geometry.

Forward velocity kinematics

The forward velocity kinematics (FVK) solves the following problem: “Given the vectors of joint positions and joint velocities, what is the resulting end effector twist?” The solution is always unique: one given set of joint positions and joint velocities always corresponds to only one single end effector twist.

Inverse position kinematics

The inverse position kinematics (IPK) solves the following problem: “Given the actual end effector pose, what are the corresponding joint positions?” In contrast to the forward problem, the solution of the inverse problem is not always unique: the same end effector pose can be reached in several configurations, corresponding to distinct joint position vectors. A 6R manipulator (a serial chain with six revolute joints) with a completely general geometric structure has sixteen different inverse kinematics solutions, found as the solutions of a sixteenth order polynomial.

Inverse velocity kinematics

Assuming that the inverse position kinematics problem has been solved for the current end effector pose, the inverse velocity kinematics (IVK) then solves the following problem: “Given the end effector twist, what is the corresponding vector of joint velocities?”

Forward force kinematics

The forward force kinematics (FFK) solves the following problem: “Given the vectors of joint force/torques, what is the resulting static wrench that the end effector exerts on the environment?” (If the end effector is rigidly fixed to a rigid environment.)

Inverse force kinematics

Assuming that the inverse position kinematics problem has been solved for the current end effector pose, the inverse force kinematics (IFK) then solves the following problem: “Given the wrench that acts on the end effector, what is the corresponding vector of joint forces/torques?”

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