Robot Arm Kinematics

Every joint of a robot arm is a rotation. Multiply the joints' matrices together, in order, and you get exactly where the gripper ends up.

gripper (1.47, 0.68, 0.83)

How it works

Matrix multiplication and rotations

In one sentence

The gripper's pose is the product of homogeneous transforms $T = R_z(\theta_1)T_z(h)R_y(\theta_2)T_x(L_1)R_y(\theta_3)T_x(L_2)$ , one per joint and link, from the base out to the tool.

General formula

{\htmlData{sym=0}{T}} = {\htmlData{sym=1}{R_z(\theta_1)}}\,{\htmlData{sym=2}{T_z(h)}}\,{\htmlData{sym=3}{R_y(\theta_2)}}\,{\htmlData{sym=4}{T_x(L_1)}}\,{\htmlData{sym=5}{R_y(\theta_3)}}\,{\htmlData{sym=6}{T_x(L_2)}}

Each matrix is one move: rotate at a joint, then slide along a link. Multiplying them stacks the moves in order, so the last column of $T$ is where the gripper ends up.

What the symbols mean

$T$
Gripper pose
All six steps multiplied together: where the gripper ends up and which way it points.
$R_z(\theta_1)$
Base swivel
Turn the whole arm by θ₁ around the vertical axis.
$T_z(h)$
Shoulder height
Move up by h, the height of the shoulder above the floor.
$R_y(\theta_2)$
Shoulder joint
Tilt the upper arm by θ₂.
$T_x(L_1)$
Upper arm
Move out along the upper arm by its length L₁.
$R_y(\theta_3)$
Elbow joint
Bend the forearm by θ₃ relative to the upper arm.
$T_x(L_2)$
Forearm
Move out along the forearm by its length L₂, arriving at the gripper.

A rotation by $\theta$ about the vertical axis is $R_z(\theta) = \begin{bmatrix}\cos\theta & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1\end{bmatrix}$ .
A $4\times4$ homogeneous matrix packs a rotation and a shift together, so one multiplication carries a point through a whole joint.
Chaining the joint matrices is forward kinematics: angles in, gripper position out.
Going backwards, from a target to angles, is inverse kinematics. For two links it reduces to the law of cosines.

Learn the math

Derivatives of Inverse Trig FunctionsCalc I · Lesson 20, Derivatives Center of MassCalc II · Lesson 13, Applications of Integrals The Cross ProductCalc III · Lesson 4, Vectors The Matrix of a Linear TransformationLin Alg · Lesson 8, Linear Equations in Linear Algebra Matrix OperationsLin Alg · Lesson 9, Matrix Algebra Matrices in Computer GraphicsLin Alg · Lesson 15, Matrix Algebra Coordinate SystemsLin Alg · Lesson 22, Vector Spaces The Dimension of a Vector SpaceLin Alg · Lesson 23, Vector Spaces Change of BasisLin Alg · Lesson 24, Vector Spaces

Learn the math

Derivatives of Inverse Trig FunctionsCalc I · Lesson 20, Derivatives

Center of MassCalc II · Lesson 13, Applications of Integrals

The Cross ProductCalc III · Lesson 4, Vectors

The Matrix of a Linear TransformationLin Alg · Lesson 8, Linear Equations in Linear Algebra

Matrix OperationsLin Alg · Lesson 9, Matrix Algebra

Matrices in Computer GraphicsLin Alg · Lesson 15, Matrix Algebra

Coordinate SystemsLin Alg · Lesson 22, Vector Spaces

The Dimension of a Vector SpaceLin Alg · Lesson 23, Vector Spaces

Change of BasisLin Alg · Lesson 24, Vector Spaces