Jacobian Matrix
A matrix that describes how the robot moves with time. Used in Optimization based IK Solvers.
Degrees of Freedom
An object in the physical world has up to 6 cartesian Degrees of Freedom (DoF). [x, y, z] and [pitch, yaw, roll].
A mechanical DoF refers to the number of points of actuation a robot has. A robot arm with 3 joints and 2 extenders has 5 degrees of freedom, for instance.
Carteisan DoF are easily limited - the end effector must exist only within this plane (limiting to just 5 degrees of freedom).
Redundancy
given: - the number of DoF in joint space - the number of DoF in task space
- - The Robot is under-actuated and the problem is kinematically deficient. The Jacobian is “wide” and the only way to solve the inverse kinematics problem is through the pseudo inverse
- - The Robot is fully actuated. The Jacobian is square and full rank meaning that the forward kinematics equation is directly invertible. The world is great!
- - The Robot is over-actuated and the problem is kinematically redundant. the only way to solve the problem is through the psuedo-inverse. The redundancy is a benefit though - it means there are more solutions for each desired pose. The human arm, for example, is over-actuated with 7 DoF (3 in the shoulder and wrist, 1 in the elbow) in a 6 DoF task space (3d real space)
Task Space
The Robot’s work space. A robot arm’s task space might be a hemisphere around its base, whereas a mobile robot’s task space might be a room or building or world.
Configuration/Joint Space
The space of all valid robot configurations. A robot configuration is defined as a vector of joint positions.
see my Textbook notes on Configuration Space
Null Space
The set of all possible solutions for some IK task. A redundant robot with 7 DoF which has 6 DoF in task space will have infinite solutions for a given end effector Pose.