Operating at force, power, and thermal limits in electrically-actuated commercial legged robots Avik De Co-founder & CTO, Ghost Robotics Previously: Postdoc @ Harvard, Ph.D. @ UPenn

Bio-inspired robot locomotion Minitaur (6kg, 8dof direct drive) 100mg 1g 10g 100g 1kg 10kg 100kg Jerboa (3kg, 4dof) Spirit 40 (12kg, 12dof) Vision 60 (43kg, 12dof)

Ghost Robotics Vision 60, Spirit 40 • Company: ~20 people, Phila., PA • Design priorities: cost- effective, efficient, sealed • Common, not bespoke, hardware components • Complexity in software, not hardware: remove sensors (e.g., force/torque) • Design optimized for mechanical, computational efficiency (higher run time, range) • Weatherproof: can use in rain and swamps • Limitations • Not payload mule (20lb advertised)

Applications: mobile sensing, mobile manipulation Persistent security with autonomous charging CBRN/EOD

Optimize design for fundamental constraints: force, power Power breakdown Power (W) Motors 180--280 Low level electronics 10 Blind locomotion <1 Gait planner <5 Autonomy 15 • “Autonomy” requires power autonomy • Maximize range before recharge • Vision 60 total CoT 0.46—0.8, depending on terrain • Also bandwidth, transparency Minitaur: update rate 1000 Hz

Motivation: analytically-guided design Motor Motor controller Gearbox Compliant element Leg kinematics Dynamic task specification for gearbox selection e.g. [De et al (2011)] e.g. [Hollerbach (1991)] [Wensing (2017)] Huge amount of past work… 𝑢, 𝑘𝑜 , … 𝑟, 𝑑, 𝑙, … , 𝐿, 𝑅, … 𝐺, 𝐽𝐺 𝑘, 𝑏, … 𝑙𝑖 , … Platform morphology 𝑑, 𝜅, 𝜌𝑡 , 𝑚𝑡 , … e.g. [De et al (2018)]

Actuators for Robotics Motor controllers • 50V single power supply • 40A+ RMS • >500A peak (voltage mode), 80A peak (current mode) • EtherCAT interface • ~200us full loop ping • Command: voltage, position, q-current • Sensor info: rotor position, q-current

Commonly-used motor models in robotics http://ctms.engin.umich.edu/CTMS/index.php?example= MotorSpeed&section=SystemModeling http://www.vgt.bme.hu/info_en/research/sim/fem/1.htm Deficiencies: • only one control input • does not explain full torque output • underestimates max power/max speed Model • 𝜄 current • 𝑣 voltage • 𝐿 inductance • 𝑅 resistance • 𝑘𝑒 back-EMF constant Problems: • Brittle (hard to generalize) • No analytical insight • Time/computation intensive Analytical tractability

Equations of motion Electrical Mechanical • Transform to rotor frame • Rotor frame EOM don’t have 𝜃𝑒 -dependence • Power constraint from electronics: A three-phase BLDC motor model https://www.mathworks.com/help/physmod/sps/ref/brushlessdcmotor.html Model • 𝜄, 𝑥 (stator, rotor) currents • 𝑣, 𝑢 (stator, rotor) voltages • 𝐿 inductance • 𝑅 resistance • 𝑘𝑒 back-EMF constant • 𝜔𝑚 mechanical speed • 𝑛 # pole pairs • 𝜃𝑒 electrical angle • 𝛽 𝜃𝑒 back-EMF waveform

Controlling the three-phase model Energy balance: Set control goal: • Minimize heat • Maximize power or torque Input electrical power Joule heating Mechanical output power Transient Conventional Strategies • Feedback torque control (TC) using current sensing • Typical PI diagonal current control • Usually set Id -> 0 • Voltage control (VC) – only control Vq

New “Angle Control” – couple d,q axes for task Peak torque Peak power Vlim- constrained • Recall current dynamics • Note that where 𝜔𝑚 - rotor speed (measurement), 𝜏𝑒 - elec time const (fixed param) • Idea: use both dq axes to align with 𝑟 when ሶ 𝑥 = 0 (tune for ҧ 𝑥 equilibrium condition) := Steady state operation: [1] A. De, A. Stewart-Height, and D. E. Koditschek, “Task-Based Control and Design of a BLDC Actuator for Robotics,” IEEE Robotics and Automation Letters, vol. 4, no. 3, pp. 2393--2400, 2019.

AC disadvantage and variation with motor design Disadvantage: heat production AC advantage (peak torque ratio) vs. VC (blue) and TC (red) [1] A. De, A. Stewart-Height, and D. E. Koditschek, “Task-Based Control and Design of a BLDC Actuator for Robotics,” IEEE Robotics and Automation Letters, vol. 4, no. 3, pp. 2393--2400, 2019.

Making it relevant to robotics 1DOF experimental setup • “Inverted hopper” with stance and aerial phases • Fully instrumented with a single actuator • Showing braking trials here 𝑚 𝐽𝑚 𝜏𝑚 𝑧 Flight Stance Lower stopping time when power-constrained [1] A. De, A. Stewart-Height, and D. E. Koditschek, “Task-Based Control and Design of a BLDC Actuator for Robotics,” IEEE Robotics and Automation Letters, vol. 4, no. 3, pp. 2393--2400, 2019.

Summary • Task-based control can help get closer to fundamental limits • Use task to inform control strategy – rethink interface to motor controllers • Co-design