PYTHON · NUMPY · IN-BROWSER

Joints & Degrees of Freedom

Define a joint between two parts and correctly state and realize its allowed motion - revolute vs. prismatic - exposing exactly one intended degree of freedom and no others.

01Challenge

Two already-correctly-placed parts: a hinge_base welded to the world and a hinge_leaf whose knuckle is concentric with the base knuckle - a Ø6 pivot line runs through both along world Y. This CAD assembly has ZERO joints: nothing yet says the parts can move relative to each other; the DOF readout reads 0 and the leaf is rigid. For the first time the Bench exposes a physics model definition beside the CAD. Your job: declare ONE joint so the leaf rotates about the pivot line (world Y) and NOTHING else can move; the pivot passes through (30,0,12). You have not been taught joint types yet - guess a hinge, a slide, or a free joint and watch what the leaf does. Common first attempts: a free joint (leaf flops in all 6 DOF - too much), a slide joint (wrong motion), or the right hinge with the wrong axis (1 0 0 instead of 0 1 0, so it folds sideways). Each failure is diagnostic.

02Model

Placing two parts touching does nothing. Either they are frozen, or - if you tell the sim they can move - they move in EVERY way. Neither is a hinge. A hinge is a promise about exactly ONE way to move. A rigid part floating in space has six degrees of freedom: three slides, three spins. 'Free' means all six. A machine is what you get when you take some away. A joint is a subtraction. A revolute - a hinge - leaves exactly one rotation about its axis and kills the other five. One degree of freedom. The axis is the whole identity of the joint: same type, different axis, completely different machine. Two joints do almost all of mechanism design. Revolute: one rotation - hinges, knuckles, wheels. Prismatic: one translation - drawers, pistons, linear slides. Both leave one DOF; the question is always rotation or translation, and about-or-along which axis. Our pivot runs along Y, so: revolute, axis zero-one-zero. Get the type right and the axis right and you have removed exactly the five degrees of freedom you meant to. That is a hinge. That is a mechanism.

A joint allows exactly its degrees of freedom.

03Guided practice

Step A (worked, full scaffold): read a known-good DIFFERENT hinge - a lid hinged about world Z through (0,0,5): type='hinge' (revolute - one rotational DOF), axis='0 0 1' (the rotation axis), pos='0 0 5' (a point the axis passes through), range='0 120' (allowed motion, degrees). Each attribute mapped: type = which DOF survives, axis = its direction, pos = where the axis sits, range = how far it may travel. Step B (fill the attributes): structure given, values blank - type='' (rotation not translation), axis=' __ __' (pivot runs along world Y), pos='30 0 12' given, range='0 90'. Hints: (1) the leaf must turn not slide - which type leaves a rotational DOF?; (2) the pivot runs along world Y - which component is 1? Expected: type='hinge' axis='0 1 0'. Step C (independent, no scaffold): the spec changes to 'the leaf must SLIDE 25 mm along the pivot line to lock, not rotate.' You must recognize this demands a prismatic (slide) joint and realize it yourself: type='slide' axis='0 1 0' range='0 25'. If you pick hinge: this spec says slide, not turn - that is a translational DOF; use a prismatic (slide) joint along the same axis. By Step C you map an English motion requirement to joint type + axis with no template.

04Feedback

PASS WHEN PASS: the leaf probes to exactly one free DOF (n_free == 1), rotational, about the target axis (axis_err <= 1.0 deg), with range 0-90 enforced - a correct revolute hinge.

FAIL: leaf moves in 6 directions - that is a free joint, not a hinge. Use type="hinge" to leave exactly 1 rotational DOF.
FAIL: leaf slides instead of rotating. Spec asked for rotation - change type from "slide" to "hinge".
FAIL: hinge axis is 90 deg off: leaf folds about X, but the pivot runs along Y. Set axis="0 1 0".
FAIL: no free DOF detected - the leaf is welded. Add a joint inside the leaf body.
FAIL: leaf rotates past 90 deg into the base (collision at 118 deg). Add range="0 90" to stop it.

05Retrieve & space

(3.1 pose) Before a joint can be correct, the pivot must be placed right - the leaf knuckle bore must be concentric with the base bore. In the CAD, what translate put it there? Re-fuses placement with jointing.
(M2.1 driven dimensions) Make the joint range a parameter swing_deg and set the leaf clearance chamfer to a formula of it. Sweep swing_deg = 90, 120 - does collision-free still hold?
(3.2 itself, spaced +1 lesson) Given a drawer that must pull out 50 mm and not rotate: name the joint type, axis, and range - pure retrieval of revolute-vs-prismatic.

06Mastery & project

Submit a hinge model that probes to exactly one free DOF (n_free == 1), of the correct kind for the given spec, about the correct axis (axis_err <= 1.0 deg), with its range enforced and no self-collision through that range - passing BOTH the revolute spec AND the prismatic spec variant. Passing both proves you can map a stated motion requirement to the correct joint type + axis and verify the DOF empirically, not just assert it. 3.3 then unlocks.

The joint is the atom of every mechanism you will simulate. 3.3 chains four of these into a linkage; M4 puts mass and torque on exactly these joints; the M5 capstone is graded on a device whose motion is only as correct as its joints. A learner who can declare 'one DOF, this axis, this range, no collision' can build any mechanism in the course.

← Locating Parts in Space A Mechanism That Moves →

0:\n axis = axis / np.linalg.norm(axis)\n trans_resp, rot_resp = [], []\n for d in TRANS_DIRS:\n if jtype == 'free':\n trans_resp.append(1.0)\n elif jtype == 'slide':\n trans_resp.append(abs(float(np.dot(d, axis)))) # only along axis\n else: # hinge / weld\n trans_resp.append(0.0)\n for d in ROT_DIRS:\n if jtype == 'free':\n rot_resp.append(1.0)\n elif jtype == 'hinge':\n rot_resp.append(abs(float(np.dot(d, axis)))) # only about axis\n else: # slide / weld\n rot_resp.append(0.0)\n return np.array(trans_resp), np.array(rot_resp), axis\n\nprint('probe() ready')","label":"Joint model - how a joint admits motion (given - do not edit)"},{"code":"# Add ONE joint so the leaf rotates about the pivot line (world Y) and NOTHING\n# else can move. Choose type ('hinge'=revolute, 'slide'=prismatic, 'free'=all 6),\n# the axis vector, and the range in degrees.\n#\n# in the full Bench this is inside .\n\njoint = {\n 'type': 'free', # <-- rotation, not translation, not all-6\n 'axis': [0, 0, 0], # <-- pivot runs along world Y\n 'pos': [30, 0, 12], # given: a point on the pivot line\n 'range': [0, 90], # degrees\n}\n\ntr, rot, axis = probe(joint)\nprint('translational response x,y,z :', np.round(tr, 3))\nprint('rotational response x,y,z :', np.round(rot, 3))","label":"YOUR JOB - declare ONE joint (edit this cell only)"},{"code":"# Sim-based DOF probe: count directions that moved, classify kind, check axis & range.\n\ntr, rot, axis = probe(joint)\nn_trans = int(np.sum(tr > MOTION_EPS))\nn_rot = int(np.sum(rot > MOTION_EPS))\nn_free = n_trans + n_rot\nkind = 'rotational' if n_rot >= n_trans else 'translational'\n\n# realized motion axis = the test direction that moved most; axis error vs target\nresp = rot if kind == 'rotational' else tr\nrealized = (ROT_DIRS if kind == 'rotational' else TRANS_DIRS)[int(np.argmax(resp))]\ncos = abs(float(np.dot(realized, TARGET_AXIS)))\naxis_err = float(np.degrees(np.arccos(np.clip(cos, -1, 1))))\n\nlo, hi = joint['range']\nrange_ok = (lo == 0 and hi == 90) # revolute spec: stop at 0 and 90 deg\n\nfails = []\nif n_free == 0:\n fails.append('FAIL: no free DOF detected - the leaf is welded. Add a joint inside the leaf body.')\nelif n_free > 1:\n fails.append(f'FAIL: leaf moves in {n_free} directions - that is a free joint, not a hinge. Use type=\"hinge\" to leave exactly 1 rotational DOF.')\nelif kind != 'rotational':\n fails.append('FAIL: leaf slides instead of rotating. Spec asked for rotation - change type from \"slide\" to \"hinge\".')\nelif axis_err > 1.0:\n fails.append(f'FAIL: hinge axis is {axis_err:.0f} deg off: leaf folds about the wrong axis, but the pivot runs along Y. Set axis=\"0 1 0\".')\nelif not range_ok:\n fails.append(f'FAIL: leaf rotates past 90 deg into the base. Add range=\"0 90\" to stop it.')\n\nprint(f'n_free={n_free} kind={kind} axis_err={axis_err:.1f}deg range=({lo},{hi})')\nif fails:\n print(fails[0])\nelse:\n print('PASS: exactly 1 rotational DOF about Y, range 0-90 enforced - that is a hinge.')","label":"Autograder - submit"}],"intro":"The real Bench probes a MuJoCo-WASM sim for DOF. Here we emulate the kinematic referee in numpy: declare a joint (type, axis, range) and the grader applies a small test wrench in all six directions, measures which produce motion, and reports n_free, the realized axis, and the kind - exactly the sim-based DOF probe, not an XML parse. Edit only the joint declaration cell, gravity off, kinematic only.","key":"design-simulation/joints-and-degrees-of-freedom","kind":"python","title":"Joints & Degrees of Freedom"}">

PYTHON · NUMPY · IN-BROWSER

Joints & Degrees of Freedom

The real Bench probes a MuJoCo-WASM sim for DOF. Here we emulate the kinematic referee in numpy: declare a joint (type, axis, range) and the grader applies a small test wrench in all six directions, measures which produce motion, and reports n_free, the realized axis, and the kind - exactly the sim-based DOF probe, not an XML parse. Edit only the joint declaration cell, gravity off, kinematic only.