✎ Modified DH Calculator
⁽ⁱ⁻¹⁾Tᵢ = Rₓ(αᵢ₋₁) · Tₓ(aᵢ₋₁) · Rz(θᵢ) · Tz(dᵢ)⌘ Kinematic Tree
DOF: 3Joint Inspector
ACTIVEEnd-effector summary
TCP| i | Joint name | aᵢ₋₁ (mm) | αᵢ₋₁ (deg) | θᵢ (deg) | dᵢ (mm) | Type | Limits | Actions |
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Robot Library
Load a starter configuration into the same Modified-DH workspace. These are educational kinematic models, not manufacturer-accurate CAD assemblies.
2R Planar Arm
Two revolute joints for basic planar forward and inverse kinematics.
3R Planar Arm
Three revolute joints for redundant planar positioning and orientation studies.
SCARA (RRP)
Two revolute joints and a prismatic vertical slide; useful for pick-and-place concepts.
4R Planar Manipulator
Four revolute joints for redundant reach and joint-space motion experiments.
5-Axis Articulated Arm
Five revolute joints with nonzero twists to explore 3D coordinate frames.
6R Industrial-Style Arm
A simplified six-revolute-joint serial arm for Jacobian, workspace and trajectory studies.
Cartesian Gantry (PPP)
Three prismatic axes with different directions; useful for studying linear stages.
Custom Arm
Start with a compact arm and modify link lengths, twists, joint types and limits.
Coordinate Frames
Frame axes are drawn from the cumulative homogeneous transformations. Red = X, green = Y, blue = Z.
Frame transforms
Forward Kinematics
Compute the base-to-tool pose by multiplying the selected Modified DH transforms in sequence.
Final transform
Inverse Kinematics
Use the analytic 2R solver for planar arms, or damped least-squares position IK for the currently loaded Modified-DH robot.
Analytical planar 2R
For a two-link planar arm. Applies joint angles to joints 1 and 2 only.
General IK · Damped least squares (position or full pose)
Verified damped-least-squares solver with multi-start, joint limits and optional orientation matching, using the analytic geometric Jacobian. Every answer is re-checked with forward kinematics; unreachable or singular targets are reported honestly.
Learning Center
Learn the parameters and understand how the transformation is built.
1. Link length · aᵢ₋₁
Translation along the previous frame's X axis. It represents the common-normal distance between adjacent joint axes.
2. Link twist · αᵢ₋₁
Rotation about the previous frame's X axis, describing the angle between the adjacent Z axes.
3. Joint angle · θᵢ
Rotation about the current joint Z axis. It is variable for revolute joints.
4. Joint offset · dᵢ
Translation along the current joint Z axis. It is variable for prismatic joints.
Your chosen transformation order
T = Rx(αᵢ₋₁) Tx(aᵢ₋₁) Rz(θᵢ) Tz(dᵢ)How to check your answer
1. Validate every frame assignment. 2. Build each 4×4 matrix. 3. Multiply left-to-right. 4. Read the final matrix's last column for position. 5. Check units and joint-axis signs.
∂ Jacobian & Velocity Analysis
Linear Jacobian Jv
Computed analytically (cross-checked against finite differences). Revolute columns are mm/rad; prismatic columns are mm/mm. Rows 1–3 are linear velocity, rows 4–6 angular velocity.
⌁ Trajectory Planner
Plan cubic or quintic joint-space motion for the selected joint. Playback updates the same robot and Modified-DH calculations used by the main workspace.
Trajectory settings
Position profile
Velocity profile
Acceleration profile
◌ Workspace Analysis
Sample the active robot's reachable TCP positions by sweeping its joint limits. Results are approximate and depend on the DH frame assignment.
⚖ Dynamics Explorer
Transparent, simplified single-joint torque estimate for learning. It is not a full multi-body inverse-dynamics solver and should not be used to size real hardware.