"""Size the legs and the drivetrain from a height budget and real parts. Answers three questions before any simulation happens: 1. How long should each leg segment be, given a target standing height? 2. What joint torque does that geometry demand? 3. What reduction, if any, do the chosen motors need to supply it? python sizing.py # the current build python sizing.py --height 0.6 --wheel 0.25 The geometry result is worth stating up front. With the hip and the wheel axle on one vertical line and two equal segments bent by theta from vertical: drop D = 2 L cos(theta) so L = D / (2 cos theta) knee load T = (W/2) L sin(theta) so T = (W D / 4) tan(theta) Knee torque depends on the bend angle and NOT on the segment length, once the height is fixed. Longer legs held straighter cost nothing extra in torque. The bend angle is the whole design variable, and it is bounded from below by the control authority and suspension travel you need, not by strength. """ import argparse import math MOTORS = { "DD-5050Pro": { "kv": 300, "v_max": 36, "v_nom": 24, "rated_a": 33, "max_a": 67, "rated_torque": 1.2, "max_torque": 2.5, "rated_w": 900, "max_w": 1800, "mass": 0.430, "size": "50x50mm", }, "DD-4125": { "kv": 300, "v_max": 25.2, "v_nom": 24, "rated_a": 12, "max_a": 18.5, "rated_torque": 8.27 / 300 * 12, "max_torque": 8.27 / 300 * 18.5, "rated_w": 178, "max_w": 178, "mass": 0.200, "size": "41x25mm", }, } # Carbon fibre tube, unidirectional-dominant layup. CF_DENSITY = 1600.0 # kg/m3 CF_E = 70e9 # Pa, conservative for a filament-wound tube def tube(od_mm, wall_mm=2.0): od, wall = od_mm / 1000, wall_mm / 1000 idm = od - 2 * wall area = math.pi / 4 * (od ** 2 - idm ** 2) I = math.pi / 64 * (od ** 4 - idm ** 4) return {"od_mm": od_mm, "wall_mm": wall_mm, "kg_per_m": area * CF_DENSITY, "area": area, "I": I, "EI": CF_E * I} def solve(height=0.60, wheel_d=0.25, chassis_h=0.12, theta_deg=25.0, tube_od=40.0, joint_motor="DD-5050Pro", wheel_motor="DD-4125", wheel_ratio=1.0, extra_mass=2.2, gearbox_mass=0.25, dynamic_factor=2.0): wheel_r = wheel_d / 2 theta = math.radians(theta_deg) # Hips sit on the chassis centre line, so the drop the legs must cover is # from there to the wheel axle. chassis_z = height - chassis_h / 2 drop = chassis_z - wheel_r if drop <= 0: raise SystemExit("wheels alone exceed the height budget") L = drop / (2 * math.cos(theta)) t = tube(tube_od) jm, wm = MOTORS[joint_motor], MOTORS[wheel_motor] seg_mass = t["kg_per_m"] * L mass = (4 * jm["mass"] + 2 * wm["mass"] + 4 * seg_mass + 2 * 0.35 + extra_mass) W = mass * 9.81 knee_static = (W * drop / 4) * math.tan(theta) hip_static = knee_static # symmetric two-link stance knee_dyn = knee_static * dynamic_factor # Wheels: what tilt can it catch? An inverted pendulum needs base # acceleration g*tan(lean) to hold a lean, so available traction sets the # largest disturbance the machine can recover from at all. wheel_tau = wm["max_torque"] * wheel_ratio traction = 2 * wheel_tau / wheel_r accel = traction / mass max_lean = math.degrees(math.atan(accel / 9.81)) no_load_rad = jm["kv"] * jm["v_nom"] * 2 * math.pi / 60 wheel_no_load = wm["kv"] * min(wm["v_max"], 24) * 2 * math.pi / 60 / wheel_ratio top_speed = wheel_no_load * wheel_r return { "height": height, "chassis_z": chassis_z, "wheel_r": wheel_r, "drop": drop, "L": L, "theta_deg": theta_deg, "mass": mass, "seg_mass": seg_mass, "tube": t, "knee_static": knee_static, "hip_static": hip_static, "knee_dyn": knee_dyn, "joint_motor": joint_motor, "wheel_motor": wheel_motor, "ratio_cont": knee_static / jm["rated_torque"], "ratio_dyn": knee_dyn / jm["max_torque"], "joint_no_load_rad": no_load_rad, "traction_n": traction, "accel": accel, "max_lean_deg": max_lean, "top_speed": top_speed, "wheel_ratio": wheel_ratio, "extend_z": wheel_r + 2 * L + chassis_h / 2, "crouch_z": wheel_r + 2 * L * math.cos(math.radians(60)) + chassis_h / 2, } def report(r): jm, wm = MOTORS[r["joint_motor"]], MOTORS[r["wheel_motor"]] print(f"\nTARGET {r['height']*1000:.0f}mm standing, {r['wheel_r']*2000:.0f}mm wheels") print("-" * 66) print(f" chassis centre {r['chassis_z']*1000:6.0f} mm") print(f" hip to wheel axle {r['drop']*1000:6.0f} mm") print(f" SEGMENT LENGTH {r['L']*1000:6.0f} mm each (thigh = shank)") print(f" total leg {r['L']*2000:6.0f} mm at {r['theta_deg']:.0f} deg bend") print(f" height range {r['crouch_z']*1000:.0f} - {r['extend_z']*1000:.0f} mm " f"(crouched to straight)") t = r["tube"] print(f"\n CF tube {t['od_mm']:.0f}mm OD x {t['wall_mm']:.0f}mm wall: " f"{t['kg_per_m']:.3f} kg/m -> {r['seg_mass']*1000:.0f} g per segment") print(f" estimated total mass {r['mass']:.2f} kg") print(f"\nJOINTS {r['joint_motor']} " f"{jm['rated_torque']:.1f} Nm rated / {jm['max_torque']:.1f} Nm peak") print(f" static hold needed {r['knee_static']:6.2f} Nm at hip and knee") print(f" with {2.0:.0f}x dynamic margin {r['knee_dyn']:6.2f} Nm") print(f" reduction for rated {r['ratio_cont']:6.2f} : 1") print(f" reduction for peak {r['ratio_dyn']:6.2f} : 1") rec = max(1.0, math.ceil(max(r["ratio_cont"], r["ratio_dyn"]))) verdict = ("DIRECT DRIVE IS ENOUGH" if rec <= 1 else f"NEEDS ABOUT {rec}:1") print(f" -> {verdict}") if rec > 1: print(f" at {rec}:1 the joint gives " f"{jm['rated_torque']*rec:.1f} Nm continuous, " f"{jm['max_torque']*rec:.1f} Nm peak, " f"{r['joint_no_load_rad']/rec:.0f} rad/s no-load") print(f"\nWHEELS {r['wheel_motor']} at {r['wheel_ratio']:.0f}:1 " f"{wm['max_torque']:.2f} Nm peak per motor") print(f" traction {r['traction_n']:6.1f} N total") print(f" base acceleration {r['accel']:6.2f} m/s2") print(f" RECOVERS LEAN UP TO {r['max_lean_deg']:6.1f} deg") print(f" top speed {r['top_speed']:6.1f} m/s " f"({r['top_speed']*3.6:.0f} km/h)") if r["max_lean_deg"] < 12: print(" -> thin. A balancer that can only catch a small lean will") print(" survive flat floors and fail on shoves, slopes and thresholds.") def main(): ap = argparse.ArgumentParser() ap.add_argument("--height", type=float, default=0.60) ap.add_argument("--wheel", type=float, default=0.25) ap.add_argument("--chassis-h", type=float, default=0.12) ap.add_argument("--theta", type=float, default=25.0) ap.add_argument("--tube", type=float, default=40.0) ap.add_argument("--wheel-ratio", type=float, default=1.0) ap.add_argument("--sweep", action="store_true") args = ap.parse_args() r = solve(args.height, args.wheel, args.chassis_h, args.theta, args.tube, wheel_ratio=args.wheel_ratio) report(r) if args.sweep: print("\n\nBEND ANGLE SWEEP (height fixed, so this is the real trade-off)") print(f" {'bend':>5} {'segment':>9} {'knee Nm':>9} {'ratio':>7} note") for th in (10, 15, 20, 25, 30, 35, 40): s = solve(args.height, args.wheel, args.chassis_h, th, args.tube, wheel_ratio=args.wheel_ratio) note = ("near-singular, poor authority" if th <= 12 else "little travel" if th <= 17 else "good compromise" if th <= 27 else "lots of travel, high torque") print(f" {th:4.0f}d {s['L']*1000:8.0f}mm {s['knee_static']:9.2f} " f"{max(s['ratio_cont'], s['ratio_dyn']):7.2f} {note}") print("\n\nWHEEL REDUCTION SWEEP") print(f" {'ratio':>6} {'traction N':>11} {'m/s2':>7} {'max lean':>9} {'top m/s':>9}") for wr in (1, 2, 3, 4, 6, 8): s = solve(args.height, args.wheel, args.chassis_h, args.theta, args.tube, wheel_ratio=wr) print(f" {wr:5.0f}:1 {s['traction_n']:11.1f} {s['accel']:7.2f} " f"{s['max_lean_deg']:8.1f}d {s['top_speed']:9.1f}") if __name__ == "__main__": main()