# Manipulation Kinematics

To control the movement of a robot arm, a mathematical representation is required to compute actuator inputs and to represent obstacles for a trajectory planner. To accomplish this, the Manipulation Kinematics GEM represents an articulated robotic system as linked rigidbodies and converts robot states between configuration space and cartesian space.

## Application Architecture

The robot arm is modeled as a tree of links, each identified by a string name. Each link can have a “motor” describing the transform between its frame and its child frame. This transform can be constant (for non-joint links and fixed joints) or parameterized for movable joints like revolute joints.

Note

Links currently do not have associated geometry (mesh) for collision modeling.

The degrees of freedom of the kinematic tree is the sum of degrees of freedom of all the motors in a kinematic chain. An example below shows a kinematic chain with four links. The links “Axis1” and “Axis2” (frame shown as solid lines) have a constant motor, and the links “Joint1” and “Joint2” (frame shown as dashed lines) have parameterized motors, with the rotation axis along (001) and (010) (in their own frames) respectively.

## Implementation Details

The Manipulation Kinematics GEM is located at //engine/gems/kinematic_tree: kinematic_tree. All calculations are performed as dual quaternions to represent displacement in a 3D space: See the //engine/core/math/dual_quaternion.hpp component for more details. The validation of kinematic states only accounts for joint limits at the moment; there are currently no other checks for self-collision.

### Forward Kinematic

To find the position of the end effector from the position of each joint, the application extracts the chain of links from the root to the end effector, multiplying the dual quaternion of each link along the way. The final dual quaternion will represent the current 3D displacement and can be converted into a Pose3d.

### Inverse Kinematic

To find a joint configuration that will bring the end effector to a given target, the application implements a gradient-descent method using Jacobian transposition. For an over-specified system–that is, one with more than six joints–this solution provides the closest configuration to the initial state provided to the solver. This s effective when the target state and current state is connected and close in the configuration pace, and when the current state is provided to the solver as an initial guess.

## Codelets

The Manipulation Kinematics GEM uses the KinematicTree codelet to load a kinematic-tree file, which has a naming structure of “<name>.kinematic.json”. The following is an example of the file format:

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{
{
"name": "base",
"motor": {
"type": "constant",
"properties": {
"pose": {
"translation": [0.0, 0.0, 0.0],
"rotation": {
"axis": [0.0, 0.0, 1.0],
}
}
}
}
},
{
"name" : "axis1",
"parent": "base",
"motor": {
"type": "constant",
"properties": {
"pose": {
"translation": [0.0, 0.0, 0.1373],
"rotation": {
"axis": [0.7071, 0.7071, 0.0],
"angle_degrees": 180.0
}
}
}
}
},
{
"name" : "joint1",
"parent": "axis1",
"motor": {
"type": "revolute",
"properties": {
"axis": [0, 0, 1]
}
}
}
]
}