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Robots move themselves (egomotion) and/or move objects around them (manipulation). Here we begin with a general framework for motion algebra. We then introduce specific representations and parameterizations of motion. These representations lead to algorithms that control motion or estimate the state of the environment from a motion model. A self-driving car and a humanoid robot may require different motion representations. The choice depends on the robot vendor or application domain (e.g., aeronautics). We use motion representations to model: Kinematics: Kinematics describes the geometric framework in which motion is represented and constrained. In a locomotive robot with a pose configuration (x,y,θ)(x,y,\theta), a kinematic model will determine how the planar coordinates (xx, yy) and its orientation θ\theta will change over time via quantities such as linear and angular velocities. This is similar in concept to the game of tennis being watched via a camera. We observe where the ball is at every frame and we can construct its trajectory without having the information of the mass of the ball or what kind of forces act on the ball (gravity, drag, etc.) Similarly for a manipulator robot we can determine the pose of its end-effector purely on the grounds of its joint coordinates and joint velocities without considering the forces or torques that cause the motion or asking the question to estimate those torques. Dynamics: Using the tennis ball analogy, we can explain why the trajectory changes when gravity acts, the racket strikes the ball, or the ball collides with the court and we need forces and masses to do so. The dynamic equations of motion relate the actuation and contact forces on a robot mechanism to the resulting acceleration and motion trajectories.

Configuration space

Degrees of freedom, joint spaces, and configuration representations.

Planar motion

Frames, SO(2) and SE(2), and planar twists: the background for wheeled robots.

Wheeled robots

Kinematic models of wheeled mobile robots, following Lynch and Park chapter 13.

Homogeneous coordinates

Representing points and transformations in projective space.

Spatial motion

SO(3), SE(3), twists and wrenches, Euler angles, and quaternions.

AR4 arm kinematics

A worked six-axis example: Denavit-Hartenberg parameters, forward and inverse kinematics, and joint angles to motor steps.

AR4 kinematics lab

FK and IK built from the as-built URDF, reconciled against the DH model.
This chapter covers the mathematical foundations of how robots are structured and move, configuration spaces, coordinate transformations, and motion models for wheeled and articulated robots.

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References

  • Modern Robotics Videos (Northwestern University), companion video lectures to Kevin Lynch and Frank Park’s Modern Robotics: Mechanics, Planning, and Control.
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