INDRBT / Reading room

Kinematic formulation

Digital engineering model. Payload and repeatability are design targets. All ordinary guide reading, figures and PDFs work here. Links marked Source file lead to the public engineering repository.

Adapted for browser reading from the project’s illustrated documentation. Numeric results and engineering scope are unchanged. Reproduction notes are optional; the robot and board viewers run from the navigation above.

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Kinematic formulation

INDRBT uses an explicit sequence of rigid-body transformations. All calculations use metres, radians and seconds. A column vector represents a point; transforms map local coordinates into the base frame. The base frame has +Z up, +X along the zero-position arm and +Y completing a right-handed frame. This is a numerical model of an original design, not a reverse-engineered commercial arm.

Frames and joint definitions

Let h, L1, L2, lr, lp and lt be the shoulder height, upper-arm, forearm, wrist-roll, wrist-pitch and tool lengths in config/robot.yaml.

Joint	Axis in its parent frame before rotation	Part frame origin	Zero convention
J1	Z	(0,0,h)	Turret aligned with base X
J2	Y	Shoulder	Upper arm points along local +X
J3	Y	Upper-arm end	Forearm points along local +X
J4	X	Forearm end	Wrist-roll reference aligned
J5	Y	Roll-section end	Wrist pitch reference aligned
J6	X	Pitch-section end	Tool roll reference aligned

Each CAD part is authored in the frame after its joint rotation and before the following +X translation. RobotArm.frames(q) returns these six transforms as joint_transforms and link_transforms (aliases), alongside joint origins, world-frame axes, link segments, identity base frame, and the final TCP transform. transforms is also an alias for those six part frames; it does not include TCP. The world joint axis is computed before applying that joint's rotation.

The final TCP follows the configured tool length. CAD and motion software use the same transform convention. The gripper's full jaw opening is along tool-frame Y, with inner jaw faces at ±opening/2 and fingers extending toward +X.

Forward kinematics

For translation T(a) and axis rotation R_i(q), homogeneous matrices are

T(a) = [ I3  a ]       R_i(q) = [ rotation_i(q)  0 ]
       [  0  1 ]                [       0        1 ]

The complete transformation is

T_base_tcp(q) = Tz(h) Rz(q1)
               Ry(q2) Tx(L1)
               Ry(q3) Tx(L2)
               Rx(q4) Tx(lr)
               Ry(q5) Tx(lp)
               Rx(q6) Tx(lt).

Positive rotation follows the right-hand rule. In particular, positive rotation about Y tips local +X downward. At all-zero angles the TCP is (L1+L2+lr+lp+lt, 0, h) with identity orientation. At q2=-π/2 and all other angles zero it is (0, 0, h+L1+L2+lr+lp+lt). Both cases are tested against exact positions. Configured joint bounds are enforced by FK unless explicitly disabled.

Inverse kinematics

The solver uses scipy.optimize.least_squares with box constraints, a supplied seed, home seed, midrange seed and deterministic random restarts. There is no analytical IK claim. The residual is

r(q) = [ (p(q)-p_target)/reach,
         log_SO3(R_target R(q)^T) ].

The SO(3) logarithm is represented as an axis-angle rotation vector. Dividing translation by the reach provides a documented dimensionless balance between position and orientation. A returned candidate is accepted only if its separate position and orientation errors satisfy the configured tolerances. These are numerical solver tolerances; they are unrelated to physical repeatability.

Targets outside the shoulder-centred maximum length bound are rejected before optimization. A failed bounded numerical search produces a useful error with the best residual. It is not a proof that a target within that bound is unreachable. Redundant branches and elbow/wrist alternatives can occur. Restarts can find an alternative branch but do not exhaustively enumerate every solution. The planner uses continuation from the preceding configuration to preserve local continuity.

Geometric Jacobian

For TCP position p, world joint origin o_i and world unit rotation axis a_i, the six columns are

J_i = [ a_i × (p-o_i) ]
      [      a_i      ].

J(q) qdot gives world-frame linear velocity followed by angular velocity. The translational rows have metre units; rotational rows are dimensionless. Singularity diagnostics divide the translational rows by the configured reach before computing singular values. This scale must be stated when comparing condition measures. Tests compare every Jacobian column with central differences of TCP position and world-frame SO(3) orientation across eight seeded poses.

Singularities and workspace

All-zero extension is singular: shoulder/elbow pitch axes become parallel and several wrist axes align. Wrist pitch near zero can align wrist-roll and tool-roll axes. Singularities reduce one or more instantaneous motion directions and can produce large joint speeds or unstable IK branches. The plotted diagnostic threshold on the smallest scaled singular value is an educational indicator, not a safety criterion or drive acceptance limit.

The workspace demonstration samples independent valid joint configurations. It produces three projections, a 3D cloud, a separate TCP-above-floor cloud, and singularity plots. The full cloud includes below-base positions allowed by the joint-limit model. TCP z>=0 filtering alone does not verify link clearance. All workspace plots omit obstacles and self-collision filtering.

The convex-hull envelope fills inaccessible holes and cannot be called actual reachable volume. Occupied voxels are a sampling proxy that depends on pitch, sample count and random seed. Orientation feasibility differs from position-only reachability. Results and their exact seeds are saved in results/workspace/metrics.json and samples.csv.

Motion planning and simulation limits

MOVE J uses monotonic linear, cubic or quintic scalar interpolation of all six joints. Polynomial velocity and acceleration maxima provide an analytic minimum duration. Linear interpolation describes a constant-velocity interior segment; instantaneous starts and stops have unbounded acceleration. It is not used in the stop-to-stop pick-and-place sequence. Quintic segments have zero endpoint velocity and acceleration, permitting continuous concatenation with dwell segments.

MOVE L follows a straight TCP line, shortest-path SO(3) SLERP and a quintic time law. Bounded IK is solved at every sample, seeded from the previous result. Large configuration jumps are rejected. Finite-difference joint velocity and acceleration are uniformly time-scaled to satisfy limits at returned samples. This is sampled verification; no between-sample bound or collision-free certificate is claimed.

The start and every solved MOVE L sample must have a smallest scaled Jacobian singular value greater than 0.001 by default; translational rows are divided by nominal reach. This numerical regularity guard rejects singular/near-singular continuation with an instruction to choose another path or use MOVE J. The threshold is configurable and is not a mechanical safety limit. A Cartesian quintic law does not guarantee bounded joint acceleration at a singular endpoint: a radial approach to full extension can have q ~ (1-u)^(3/2) and divergent joint acceleration. Endpoint derivative assignments therefore require a regular Jacobian. Between-sample singularity and collision certification remain absent.

The FK animation evaluates the transform chain at each interpolated joint pose. It uses stick geometry and a scripted object-attachment event. It is an ideal kinematic simulation, without contact, rigid-body payload dynamics or collision physics. The PID simulation separately tests assumed independent inertia/damping joint plants with torque saturation and conditional-integration antiwindup.