3D robot arm simulator

Define the joints of a serial robot with the Denavit–Hartenberg table and see it in 3D. It computes forward kinematics with homogeneous transformation matrices, inverse kinematics to follow lines and circles, the Jacobian with the end-effector velocities and every motor torque from the dynamics.

What you can do

SCARA, articulated and Cartesian robots as examples.
Transformation matrices step by step.
Workspace and end-effector trajectories.
Actuator torques and forces with payload and gravity.

Frequently asked questions

What are Denavit–Hartenberg parameters?

Four values per joint (θ, d, a, α) that describe how to go from one reference frame to the next.

What is a robot Jacobian?

The matrix that maps joint velocities to the linear and angular velocity of the end effector.

Guide: robot arm kinematics

A robot arm is a chain of links connected by revolute (R) or prismatic (P) joints. Forward kinematics answers where the tool is for given joint angles; inverse kinematics, which angles are needed to bring it to a point. This guide explains the Denavit–Hartenberg convention, both kinematics and the Jacobian, with a verified example.

The Denavit–Hartenberg convention

Each joint i is described by four parameters: θ_i (rotation about z), d_i (offset along z), a_i (length along x) and α_i (twist about x). With them, the transformation from one frame to the next is

A_i = Rot_z(θ_i) · Trans_z(d_i) · Trans_x(a_i) · Rot_x(α_i)

For a revolute joint the variable is added to θ; for a prismatic one, to d. Multiplying them all, T = A₁·A₂·…·A_n gives the tool position (fourth column) and its orientation (the first three columns).

Example: 3-joint anthropomorphic arm

A rotating base (q₁) with the shoulder 0.4 m high (d₁ = 0.4 m, α₁ = 90°), a 0.5 m upper arm (q₂) and a 0.4 m forearm (q₃). It is the robot the tool opens with.

Forward kinematics with q = (0°, 40°, −70°): with q₁ = 0 the arm works in the x–z plane. The forearm sits at 40° − 70° = −30° from horizontal, so x = 0.5 cos 40° + 0.4 cos(−30°) ≈ 0.383 + 0.346 = 0.729 m and z = 0.4 + 0.5 sin 40° + 0.4 sin(−30°) ≈ 0.4 + 0.321 − 0.2 = 0.521 m. The tool is at (0.729, 0, 0.521) m.
Inverse kinematics to get back to that point: relative to the shoulder, the horizontal distance is 0.729 m and the vertical 0.121 m. By the law of cosines, cos q₃ = (0.729² + 0.121² − 0.5² − 0.4²)/(2·0.5·0.4) ≈ 0.342, so q₃ = ±70°.
Two solutions: with q₃ = −70° you get q₂ = 40° (elbow up), and with q₃ = +70°, q₂ ≈ −21.1° (elbow down). Both put the tool at the same point, but the second is outside this robot's shoulder limit (q₂ ≥ −10°), so only the first is reachable.

The tool gives the same position with those angles, and its numerical inverse kinematics recovers q₂ = 40° and q₃ = −70°.

The Jacobian

The Jacobian J relates the joint velocities to the tool velocity: v = J_v·q̇. For a revolute joint, its column is z_i−1 × (p − o_i−1); for a prismatic one, z_i−1. Where J loses rank, the robot is at a singularity: it loses the ability to move in some direction, as with the arm fully stretched out. Manipulability, √det(J_vJ_vᵀ), measures how far you are from a singularity.

How to use the tool

Load a robot (anthropomorphic 3R, PUMA-type or collaborative 6R, SCARA, Cartesian, planar 2R) or define your own Denavit–Hartenberg table.
Set the initial position and limits of each joint, and choose the motion: oscillate the joints, point to point, straight line or circle of the tool.
Read the tool's position and orientation, the A_i and T matrices step by step, the Jacobian, the velocity, the manipulability and the motor torques with the masses and payload.

Common mistakes

Multiplying the matrices in another order: matrix multiplication is not commutative.
Mixing degrees and radians in the Jacobian velocities.
Keeping only one inverse kinematics solution without checking the others or the joint limits.
Getting close to a singularity: the required joint velocities blow up.

More tools

Versión en español

About Simuveliq · Privacy policy · Contact