INDRBT / Reading room

Understanding the INDRBT digital robot

This guide follows one engineering question through the project: can the chosen geometry move a tool through a useful task, support an assumed load, and be controlled in a clearly defined digital model? Read it alongside the requirements, interactive CAD twin and engineering report. The current numerical values live in the linked result files and KPI table, so this guide concentrates on the equations and how to interpret their evidence.

For optional inspection in a CAD or electronics application, use the public inspection releaseSource file and native-file opening guide. Ordinary browser inspection remains available through the showcase. The published GitHub Pages showcase includes the browser viewers and download controls.

1. Start with units and assumptions

The configuration is the common source for geometry, joint bounds, assumed masses, motion limits, gripper parameters and controller gains. Calculations use metres, radians, kilograms, seconds and newtons. CAD converts metres to millimetres at its export boundary. Degree-valued input fields are converted to radians when the software loads them.

Joint inertia and beam second moment are different physical quantities despite often sharing the letter I. The former describes resistance to angular acceleration; the latter describes a section's resistance to bending.

Payload, material properties, COM locations, friction, inertia and damping are design inputs or assumptions. FK reconstruction and native CAD/PCB checks are digital evidence. Physical payload and repeatability remain targets. These categories are carried through the scope audit.

2. Turn joint angles into the tool pose

The base frame has Z upward and X along the extended arm. Positive rotations use the right-hand rule; positive Y rotation tips an initially horizontal link downward. A homogeneous transform maps a local point into the base frame using column vectors. Its rotation block gives orientation and its final column gives position.

The serial chain is:

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

The local joint axes are Z, Y, Y, X, Y, X. Each CAD component belongs to the frame after its joint rotation and before the next translation. This convention makes the CAD assembly and the mathematical arm share the same joint origins.

With every angle zero, the TCP lies at (R, 0, h), where R = L1 + L2 + lr + lp + lt. With only q2 at −π/2, it lies at (0, 0, h+R). These are useful hand checks for signs, units and translation order. R is a shoulder-centred radial bound; h+R is a vertical height from the base plane. The kinematic formulation explains every frame and the exact software interface.

3. Solve a target pose and understand singularities

Inverse kinematics asks for joint angles that reconstruct a requested position and orientation. The bounded numerical solver minimizes:

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

The SO(3) logarithm is a three-component axis-angle rotation vector. Scaling position by R balances dimensionless position and orientation residuals. Joint bounds constrain the search, and separate configured position/orientation tolerances decide whether a candidate is accepted. Deterministic restarts improve the chance of finding a branch. A failed local search inside the radial bound does not prove that every possible branch is inaccessible.

The geometric Jacobian maps joint speed into world-frame TCP linear and angular velocity. For joint axis a_i through origin o_i, its column is:

J_i = [a_i × (p_tcp − o_i); a_i]
[v_tcp; omega_tcp] = J(q) qdot

A singularity removes an instantaneous motion direction. Near one, a modest tool motion may need large joint speeds or cause an IK branch jump. The plotted singular values use translational rows divided by R; their numerical threshold is a diagnostic for this chosen scaling. It is not a safety limit. Numerical validation records perturbed-seed reconstruction and central-difference Jacobian checks. An IK residual measures agreement between two mathematical poses; it does not measure physical accuracy or repeatability.

4. Plan smooth motion

For a joint move, every joint uses q(t) = q_start + delta_q s(u), with u = t/T. The three interpolation laws are:

For positive speed/acceleration limits, the minimum duration follows directly:

T >= max over joints of {
     velocity_factor · |delta_q| / velocity_limit,
     sqrt(acceleration_factor · |delta_q| / acceleration_limit)
     }

Cubic motion has zero endpoint velocity. Quintic motion also has zero endpoint acceleration, making it suitable for joining rest-to-rest moves and dwells. Linear interpolation is used for a comparison of the constant-speed interior; it does not meet finite start/stop acceleration requirements.

MOVE J applies these laws in joint space, so its TCP path generally curves. MOVE L uses a straight position line, shortest-path orientation SLERP and a quintic time law. It solves IK successively from the preceding joint solution, rejects large branch jumps, estimates derivatives by finite differences, and uniformly lengthens time to obey their sampled limits. This does not establish derivative bounds between samples. Joint bounds and IK success also do not establish collision clearance.

The default MOVE L regularity guard requires the smallest scaled Jacobian singular value to exceed 0.001 at the start and solved samples, with translation rows divided by reach. Singular continuation raises a useful error and suggests another path or MOVE J. Cartesian quintic endpoints alone do not establish finite joint acceleration at full extension; the inverse map can be singular. This is a sampled numerical guard with a configurable threshold, not a safety claim.

Trajectory evidence and the recorded task expose the actual samples. The object follows a scripted attachment between close/open events; grasp contact and friction are not simulated. Separate CAD clearance evidence screens all component pairs at representative task configurations, with its finite sampling scope stated explicitly.

5. Calculate gravity and choose a conceptual drive envelope

Each moving group has an assumed midpoint COM, and the payload acts at the TCP. For mass m, gravity force is [0, 0, −m g]. Its external joint moment is:

tau_gravity_i = a_i · [(p_COM − o_i) × F_gravity]
tau_holding_i = −tau_gravity_i

Only downstream masses contribute to a joint. The negative potential-energy gradient supplies an independent check on the sign and magnitude. Full extension also permits hand calculations using horizontal lever lengths. Centred lumps give zero gravity moment around the base yaw and tool roll axes; they still need torque for acceleration and damping.

The controller's inertia is an independent educational plant assumption. It must not automatically become the actuator-sizing inertia: a point payload alone has rotational inertia m · r_perpendicular², which can exceed that assumption. For each joint, the load analysis bounds the perpendicular lever to every downstream COM point using the actual link lengths. It then calculates:

I_point_bound_i = sum over downstream COM points of (m · lever_bound_i²)
I_sizing_i = max(I_control_i, I_point_bound_i)

Roll-axis offsets along the same axis contribute no perpendicular lever. The point-mass bound covers every joint angle under the midpoint COM model. An independent test differentiates COM positions at unit joint speed and checks 2 · kinetic_energy = I_ii against these bounds. The configured control plants remain unchanged. Conceptual sizing adds the gravity bound to this equivalent diagonal inertia allowance and the assumed damping allowance:

tau_output = safety_factor · (gravity_bound + I_sizing·amax + b·vmax)
tau_motor = tau_output / (gear_ratio · efficiency)

The simultaneous maxima define an envelope for this independent-axis point-mass model. They do not bound a coupled robot's complete dynamic torque: centroidal body rotation tensors, off-diagonal inertia, Coriolis/centrifugal terms and reflected drive inertia are absent. Continuous heating, brakes, gear/bearing loads, backlash and commercial part ratings also require further engineering. Load results and mechanical analysis retain the assumptions, sampled maxima and all-angle bounds separately.

6. Check the opposed-screw gripper

The inner jaw faces lie at ±opening/2. Both nuts travel one screw lead per revolution in opposite directions, so a revolution changes the full gap by twice the lead. For the sum N_total of both contact normal forces:

gap_change = 2 · lead · screw_revolutions
input_work_per_revolution = 2π · torque
output_work_per_revolution = N_total · lead
required_torque = N_total · lead / (2π · efficiency)
friction_payload_bound = mu · N_total / (grip_safety_factor · g)

Summing both jaws' work avoids a factor-of-two error. The friction bound assumes a vertical, static load, the stated coefficient and shared contacts; acceleration, contact pressure, object damage and guide friction remain unmodelled. Screw efficiency is an assumption and does not establish self-locking or brake holding. The rotary-to-linear torque convention agrees with the Thomson engineering catalogue, page 49. Current calculations are in the gripper section of the mechanical analysis.

7. Estimate beam stiffness and nominal stress

The links are ideal clamped hollow rectangular beams. For width B, bending depth H and wall t, the section model is:

I_beam = [B H³ − (B−2t)(H−2t)³] / 12
M_root = F L + M_tip + w L²/2
sigma_root = M_root H / (2 I_beam)
delta_tip = [F L³/3 + M_tip L²/2 + w L⁴/8] / (E I_beam)
theta_tip = [F L²/2 + M_tip L + w L³/6] / (E I_beam)

F is an end force, M_tip an end couple and w a distributed load in N/m. Superposition follows the small-deflection moment-curvature relation taught in MIT's cantilever lecture. The complete extended-pose TCP estimate must carry both beam-end rotations across their downstream rigid offsets:

delta_TCP = delta_upper + delta_forearm
          + theta_upper   · (L2 + lr + lp + lt)
          + theta_forearm · (lr + lp + lt)

The final theory audit added the forearm-rotation term omitted from the earlier estimate. An independent regression integrates the load moment multiplied by the unit-TCP-load moment over both flexible sections. It agrees with the corrected four-term expression. Structural results record each contribution, slope and the total, making the correction inspectable.

Wrist/tool sections are rigid in this calculation. Holes, the terminal clevis, contacts, gears, bearings, shear deformation and fatigue are omitted. Yield divided by nominal section stress describes this ideal section alone. Static deflection describes a load-dependent pose bias. Repeatability describes the spread of repeated returns under defined conditions; this project supplies no physical repeatability test.

8. Understand the controller experiment

Each joint follows an independent assumed plant and a saturated feedback command:

I_joint qddot + b qdot = tau_command
error = q_reference − q_actual
tau_unsaturated = Kp·error + Ki·integral(error) + Kd·(qdot_reference − qdot_actual)
tau_command = clip(tau_unsaturated, −torque_limit, +torque_limit)

Kp has units N·m/rad, Ki N·m/(rad·s), and Kd N·m·s/rad. Conditional integration freezes the integrator when saturation would be pushed further and permits integration that helps unwind it. Optional feedforward adds I_joint qddot_reference + b qdot_reference; the retained main control experiment uses feedback alone.

Semi-implicit Euler first updates velocity and then position, using a timestep no greater than the configured integration interval. Positions clip at joint bounds and clipped joints reset velocity; that simple boundary rule is not contact or impact dynamics. Gravity, cross-axis coupling, sensor noise, backlash and drive delay are absent. The planned CAD animation and the feedback tracking experiment are separate outputs.

Control results report joint-space RMSE, final error, command peaks and step settling. The two-percent settling time is the first sample after the last excursion outside the band, rather than the first crossing. These metrics describe the assumed plant; they do not measure physical bandwidth or TCP repeatability.

9. Read the research plots as controlled numerical experiments

Experiment data records the samples and interpretations used to generate every plot. Workspace hulls fill inaccessible holes, and occupied voxels depend on sampling density and pitch. Neither is a certified reachable or collision-free volume. Higher PID gain can reduce RMSE without improving the two-percent settling time; the measured criterion matters.

10. Follow each claim to its evidence

Use the KPI table for current values, the requirements matrix for acceptance, the completion checklist for deliverables and publication, and the final theory review for the independent cross-checks. The electronics guide covers the native schematic/PCB, power calculations and enabled-rule checks. Digital ERC/DRC verifies specified rules and connectivity; it does not establish thermal, EMC, ESD or functional-safety performance.

From the source root, python scripts/run_validation.py --help describes the full validation workflow. python scripts/review_theory.py --help describes the focused theory checks. Passing evidence applies to the documented models and samples. It should always be read together with their assumptions and limits.