[Robot Hardware 07] - Actuators (6): Heat
Temperature dependence of torque constant and resistance, efficiency maps, and continuous torque limits of QDD actuators
A motion that worked well for 30 seconds on a cold robot can become shaky after 10 minutes. The policy is the same, the joint angles are the same, and the torque commands are the same, but the foot starts slipping, the posture sags, and the actuator reaches its current limit more often. Without looking at temperature in the logs, this can look as if the controller suddenly became worse.
But a heated actuator is not the same plant it was at the beginning. The resistance of the copper windings increases, the permanent-magnet flux changes, and the loss characteristics of the inverter and bearings change as well. The current controller can hide some of these effects until voltage saturation or thermal derating begins. After that, the relationship between the torque command and the actual output can break down quickly.
$K_t$ Is Not a Single Number Engraved on the Motor
Motor datasheets list a torque constant $K_t$ in $\mathrm{Nm/A}$. For an ideal SPMSM, $K_t$ is determined by the magnet flux and the number of pole pairs, so we can write
\[\tau_e=K_t i_q\]This equation is extremely useful for actuator design, but it does not mean that $K_t$ is constant under every operating condition.
- As magnet temperature increases, remanence and flux linkage change.
- Large q-axis and d-axis currents produce magnetic saturation and cross-saturation.
- Manufacturing variation, air-gap variation, and magnet assembly errors change $K_t$ from one motor to another.
- If the encoder electrical angle is wrong, some of the measured q-axis current is actually applied along the d-axis.
- The numerical interpretation changes depending on whether the datasheet defines current as phase peak or RMS current.
The actual torque map is therefore closer to the following function than to one constant:
\[\tau_e=f(i_d,i_q,\theta_e,T_m)\]Here, $T_m$ is the rotor or magnet temperature. It is important to distinguish winding temperature $T_w$ from magnet temperature. They are connected through thermal resistances and capacitances, but they do not have to be at the same temperature at the same time. Assuming that one stator thermistor directly represents the instantaneous rotor-magnet temperature can produce significant error under a fast duty cycle.[1,2]
Within the normal operating-temperature range, the reduction in magnet flux caused by temperature is generally reversible. As the magnet cools, much of its flux returns. If the allowable temperature is exceeded, however, or if a strong opposing magnetic field is present at the same time, irreversible demagnetization can occur. In that case, cooling does not restore the original $K_t$. Ordinary thermal drift and irreversible demagnetization should therefore not be treated as the same phenomenon.
Resistance Moves First as Temperature Rises
The resistance of a copper winding increases approximately with temperature:
\[R(T_w)=R_0\left[1+\alpha_{Cu}(T_w-T_0)\right]\]The temperature coefficient of copper is usually close to $\alpha_{Cu}\approx0.0039/\mathrm{K}$.[1] If a winding rises from 25°C to 75°C, this simple approximation gives an approximately 19.5% increase in resistance. Three-phase copper loss is fundamentally $I^2R$, expressed using the appropriate peak or RMS convention.
\[P_{Cu}\propto I_{\mathrm{rms}}^2R(T_w)\]For example, using phase RMS current and phase resistance gives $P_{Cu}=3I_{\mathrm{phase,rms}}^2R_{\mathrm{phase}}$. With an amplitude-invariant dq convention, the expression is $P_{Cu}=\frac{3}{2}R_s(i_d^2+i_q^2)$. The transient figure above folds this coefficient into one equivalent $R$ in a simplified model, so the current and resistance definitions must be matched before comparing the model with a datasheet or measurement.
This is where a common misunderstanding appears. An increase in resistance does not immediately make the current controller lose the same proportion of torque. If sufficient bus voltage is available, the current loop can apply more voltage and maintain the same current. The costs are increased loss and reduced voltage margin.
The q-axis voltage equation of a PMSM can be written as
\[v_q =R_s i_q +L_q\frac{di_q}{dt} +\omega_e\left(L_d i_d+\lambda_m\right)\]For typical steady-state operation, with $i_d\approx0$ and $di_q/dt\approx0$, this simplifies to $v_q\approx R_s i_q+\omega_e\lambda_m$. As temperature increases and the $R_s i_q$ term grows, more voltage is required to maintain the same current. If the motor is also running fast, the back-EMF term is already large, so the inverter reaches its voltage limit earlier. From that point on, the current loop can no longer track the reference current and the actual torque falls.
Temperature Rise Can Reinforce Losses
An example from a manufacturer shows a neodymium-magnet motor whose torque constant decreases by about 5% after a 50°C temperature increase, while resistance increases according to the copper temperature coefficient.[1] This value is not a universal law for every motor, but it illustrates the direction of the two effects acting together.
Consider a simplified example in which the temperature rises from 25°C to 75°C:
\[\frac{R_{hot}}{R_{cold}}\approx1.195,\qquad \frac{K_{t,hot}}{K_{t,cold}}\approx0.95\]To maintain the same torque, the current must increase by approximately $1/0.95$. The ratio of copper losses is then
\[\frac{P_{Cu,hot}}{P_{Cu,cold}} \approx \frac{1.195}{0.95^2} \approx1.32\]Under these example conditions, the copper loss required to maintain the same torque increases by about 32%. As temperature rises, resistance and $K_t$ become less favorable; producing more loss to maintain the same torque makes the actuator hotter again. This creates a reinforcing effect. However, sufficient cooling or active derating can lead to a new, higher equilibrium temperature, so this does not always mean thermal runaway. The actual values depend on magnet grade, winding design, cooling, current convention, and saturation, and must be checked with a hot test of the specific motor.
Are Motors Really More Efficient at Higher Speeds?
Efficiency must be defined according to the direction of energy flow. In the motoring region,
\[\eta_{\mathrm{mot}} =\frac{P_{\mathrm{mech,out}}}{P_{\mathrm{elec,in}}}\]In the regenerative-braking region,
\[\eta_{\mathrm{regen}} =\frac{\left|P_{\mathrm{elec,returned}}\right|} {\left|P_{\mathrm{mech,absorbed}}\right|}\]At stall, $\omega=0$, so mechanical output power is zero even when the joint is supporting a large load. Current still flows and copper loss continues. This is why a static squat, holding a heavy arm horizontally, or pushing against the ground for a long time is thermally expensive.
As speed increases from zero, mechanical output $\tau\omega$ appears even at the same torque, so there is often a region in which efficiency improves. This is the background behind the statement that “motors are efficient at high speed.” Efficiency does not improve monotonically forever, however. At high speed, iron loss, eddy-current loss, windage, bearing loss, and inverter switching loss increase, while back-EMF brings the motor closer to its voltage limit. The efficiency peak is usually inside the torque–speed plane.[1]
A robot’s duty cycle is different from that of a fan or pump. Rather than rotating at a constant speed and direction for a long time, a robot accelerates in both directions, stops to hold torque, absorbs collision energy, and accelerates in the opposite direction again. At every direction reversal, speed passes through zero, but the current needed to resist inertia, gravity, or contact forces may not be zero. Efficiency is structurally low in these intervals.
Braking energy can be regenerated, but “negative torque means no loss” is also wrong. The bus and battery must be able to accept the energy, and the motor windings and inverter still have conduction and switching losses. If the battery is full or the charging-power limit is reached, a brake resistor or another energy path is required.
Why Low Reduction Ratios Are Thermally Disadvantaged—and Why We Still Use QDD
For the same motor and an ideal reducer, joint torque is approximately
\[\tau_j\approx\eta_g N K_t i_q\]so the required current is
\[i_q\approx\frac{\tau_j}{\eta_g N K_t}\]At the same joint torque, lowering the reduction ratio $N$ means that the motor must produce more torque and current directly. For the same motor, copper loss becomes unfavorable approximately as
\[P_{Cu}\propto \frac{\tau_j^2R} {\eta_g^2N^2K_t^2}\]At the same joint speed, motor speed is $\omega_m\approx N\omega_j$. A lower ratio therefore also lowers motor speed, which can make it harder to use the motor’s most efficient operating region.
On the other hand, a high reduction ratio increases reflected motor inertia as
\[J_{ref}=N^2J_m\]and makes friction and hysteresis from seals, bearings, and gear mesh appear more strongly at the joint. External-force transparency, backdrivability, and contact bandwidth can suffer as a result.[3,4] QDD gives up some of the torque multiplication provided by the reducer in order to reduce this distortion.
It is therefore only half right to conclude that “QDD is inefficient” or that “a low reduction ratio always causes more heat.” With the same motor size and the same motor, the reducer provides less thermal margin at a low ratio. In practice, QDD designs optimize a large air-gap radius, high copper fill, short end turns, direct heat transfer to the housing, forced-air or liquid cooling, and an inverter that can support regeneration. The MIT Cheetah family of designs also treated low inertia, torque density, and thermal behavior together at the motor and transmission levels.[3–5,7]
What QDD gains is not free efficiency, but external-force transparency, low reflected inertia, and controllable contact. What it gives up is the torque multiplication provided by the gears and some thermal margin for static loads. The important side depends on the robot’s task and duty cycle. QDD is also used in systems where interaction with people matters, such as exoskeletons, but continuous thermal envelope—not only peak torque—is a central design variable.[6]
The Optimal Reduction Ratio Cannot Be Chosen from Copper Loss Alone
The relationship between joint speed and motor speed is
\[\omega_m=N\omega_j\]Increasing the reduction ratio decreases the motor current and copper loss required for the same joint torque, but speed-dependent iron loss, friction loss, windage, and voltage requirements increase with motor speed. The reducer’s own friction and transmission loss also depend on reduction ratio and operating point. These effects can be grouped conceptually as
\[P_{\mathrm{loss}}(N) \approx \frac{A}{N^2} +P_{\mathrm{speed}}(N\omega_j) +P_{\mathrm{gear}}(N,\tau_j,\omega_j)\]If only the first term is considered, a high reduction ratio is always advantageous. Once speed-dependent loss, reducer loss, reflected inertia, backlash, and backdrivability are included, the optimum depends on the duty cycle. QDD is not a design that follows only this thermal optimum; it is a system-level trade-off that accepts some additional copper loss to obtain external-force transparency and impact tolerance.
Look at RMS Current and Thermal Paths, Not Only Peak Torque
The simplest lumped thermal model is
\[C_{th}\frac{dT}{dt} =P_{loss} -\frac{T-T_{amb}}{R_{th}}\]$C_{th}$ is thermal capacitance, so it helps the actuator tolerate a short peak. $R_{th}$ describes the ability to reject heat to the surroundings and determines the long-term temperature rise. Two actuators with the same peak torque can have very different continuous torque because of motor mass, winding-to-stator contact, potting, housing area, and airflow inside the joint.
This one-node model is useful for quickly estimating total temperature rise, but it cannot represent the different time constants of the winding and housing. With at least two thermal nodes, we can write
\[C_w\dot T_w =P_{Cu} -\frac{T_w-T_h}{R_{wh}}\] \[C_h\dot T_h =\frac{T_w-T_h}{R_{wh}} +P_{\mathrm{other}} -\frac{T_h-T_{\mathrm{amb}}}{R_{ha}}\]Adding rotor-magnet, stator, inverter, and reducer nodes can represent internal temperature differences more accurately. The trade-off is that more thermal resistances and capacitances then have to be identified.
When comparing actuators, we should therefore consider all of the following together:
- cold and hot $K_t$ and phase resistance
- allowable peak current and its duration, including the initial temperature
- thermal time constants of the winding, stator, housing, and magnet
- continuous torque under specified ambient and cooling conditions
- RMS current and the speed–torque histogram of the actual trajectory
- regenerative, voltage, and temperature limits of the inverter and battery
It is easy to hit peak torque once in a short bench test. It is much harder, and much more important, to verify that torque-tracking bandwidth and contact behavior remain stable after the robot has followed the same policy for 30 minutes.
Lookup Tables Can Compensate, but They Cannot Fully Observe the Thermal State
If torque constant and resistance change with temperature, we can measure the actuator at multiple temperatures and operating points and build a lookup table.
For example, an experimentally identified map could be written as
\[\hat{\tau} =f_{\mathrm{LUT}}\left(i_q,\omega_m,T_s,V_{\mathrm{bus}}\right)\]Here, $T_s$ is the measurement from a stator- or housing-mounted temperature sensor. Measuring actual output torque at different currents, speeds, and temperatures makes it possible to use an effective torque constant that varies with operating condition instead of one fixed $K_t$.
\[\hat K_t=K_t(i_q,\omega_m,T_s)\]This compensation is useful for reducing errors caused by flux variation, magnetic saturation, inverter voltage drop, and repeated current-to-torque mismatch. But its accuracy depends on the assumption that the table inputs sufficiently represent the internal state of the actuator. The biggest problem is that the permanent magnets that directly affect torque constant are located on the rotating rotor.
Rotor Temperature Is Hard to Measure Directly
The winding and stator are stationary, so a thermistor, RTD, or thermocouple can be attached relatively easily. The permanent magnets and rotor core, on the other hand, continue to rotate. Even if a sensor is attached to the rotating component, a slip ring, wireless telemetry, or dedicated electronics on the rotor are needed to transfer power and signals to the outside.
These structures add several burdens to a typical robot actuator:
- increased rotating mass and inertia
- more complicated wiring and packaging
- reliability issues in wireless communication and power delivery
- mechanical retention problems at high rotational speed
- additional cost and failure modes
For this reason, real actuators often estimate rotor-magnet temperature indirectly using stator temperature and electrical signals instead of measuring it directly.
\[\hat T_m =g\left(T_s,i_d,i_q,\omega_m,V_{\mathrm{bus}},t\right)\]For example, changes in back-EMF or flux linkage can be used to estimate magnet temperature, while changes in winding resistance can be used to estimate winding temperature. These estimates depend on motor parameters, current-measurement accuracy, electrical-angle error, and operating history.
The winding, stator, housing, and rotor also have different thermal time constants. Immediately after a large current is applied, winding temperature can rise first, while the rotor magnet and housing respond more slowly. Thus, even when a housing thermistor reports the same temperature, the internal winding and magnet temperatures can differ depending on the previous operating history.
In other words, a single measurable $T_s$ can correspond to multiple internal thermal states.
\[T_s \;\not\Rightarrow\; \left(T_w,T_m\right) \text{ is uniquely determined}\]This is why a temperature-axis lookup table cannot completely compensate for every kind of thermal drift. In addition to the current sensor value, the table may need current and speed history, cooling conditions, or the state of a thermal observer.
An Actuator Identified on the Bench Becomes a Different Thermal System on the Robot
Even after thorough system identification of a standalone actuator on a test rig, the thermal boundary conditions change as soon as the actuator is mounted on the robot.
Heat generated in the motor moves from the winding to the stator, housing, robot link, and surrounding air. The actuator’s thermal behavior is therefore not determined by motor parameters alone.
\[T =f\left( P_{\mathrm{loss}},R_{\mathrm{internal}},C_{\mathrm{internal}}, \mathrm{mounting},\mathrm{housing},\mathrm{airflow},T_{\mathrm{amb}} \right)\]Even with the same motor and reducer, the following conditions can change thermal resistance and thermal time constants:
- housing material, thickness, and surface area
- contact area and thermal interface material between the motor and housing
- bolt preload and assembly tolerances
- conductive heat paths into the links
- whether the joint interior is sealed
- natural convection, fans, forced air, or liquid cooling
- heat arriving from neighboring joints and electronic components
- airflow and natural convection determined by robot posture
- additional losses in reducer lubricant and bearings
On a bench, the actuator may be firmly attached to a large aluminum fixture that acts as a heat sink. In the actual robot, it may instead be enclosed in a plastic cover or a thin link with little airflow, making heat removal slower. The opposite can also happen: a standalone test may use only natural convection, while the complete robot link becomes a large heat-spreading structure after installation.
The $R_{th}$ and $C_{th}$ identified in a standalone test are therefore not absolute constants intrinsic to the actuator. They are closer to equivalent parameters measured under a particular mounting and cooling condition.
\[R_{th}=R_{th}(\mathrm{installation})\] \[C_{th}=C_{th}(\mathrm{installation})\]Changing the housing does not merely change the cooling rate. Housing stiffness, bearing alignment, preload, and lubrication can also change friction loss and transmission efficiency. A mechanical-design change can therefore modify both the thermal model and the torque-transmission model.
System Identification Does Not End with a One-Time Calibration
For this reason, actuator system identification is more practical when divided into two stages.
The first stage identifies characteristics of the actuator itself that are relatively portable:
- motor torque as a function of current
- electrical-angle and current-sensor errors
- temperature dependence of winding resistance
- temperature dependence of flux linkage and torque constant
- voltage and current limits of the motor and inverter
- reducer friction and transmission efficiency
The second stage re-identifies system-level characteristics with the actuator installed on the actual robot:
- thermal resistance and thermal time constants in the installed configuration
- heat transfer through the actual links and housing
- cooling conditions at each joint
- temperature changes caused by robot posture and motion
- thermal derating during continuous operation
- thermal coupling between neighboring joints
Standalone bench testing is necessary, but it is not sufficient. The plant that ultimately interacts with the policy and controller is not just the motor and reducer. It is the complete actuator system, including the housing, links, power supply, cooling, and operating history.
Lookup tables must also be validated again after installation on the robot. Instead of using a bench-generated map unchanged, it is safer to use the standalone results as an initial model and update the thermal model and available-torque estimate using logs from the real robot.
The goal of compensation is not to create one perfect torque map for every condition. It is to estimate the uncertainty in the current internal thermal state and provide a safe torque range together with model confidence.
\[\hat{\tau}_{\mathrm{available}}, \qquad \sigma_{\tau}, \qquad \hat T_w, \qquad \hat T_m\]Heat Is a Slow State the Policy Cannot Observe
A simulator’s actuator usually has the same $K_t$, resistance, and torque limit at the beginning and end of an episode. On a real robot, the motion in the previous episode changes the plant for the next one.
\[\tau_{actual} =f\!\left( u,\omega,T_w,T_m,V_{bus} \right)\]If temperature is not included in the observation, the policy cannot tell why the same state and action produce a different acceleration. Once a current limit or thermal derating becomes active, the change can also be discontinuous. If one joint heats up before the others, left–right symmetry is broken as well.
The high-level controller or policy does not need to receive every internal temperature directly. The low-level actuator controller can combine thermal, voltage, and regenerative limits to compute the currently available torque.
\[\tau_{\mathrm{available}} =f\left(T_w,T_m,T_{\mathrm{inv}},V_{\mathrm{bus}},\omega\right)\]This interface should ideally include available torque in both the positive and negative directions, thermal headroom, derating status, voltage margin, and available regenerative power. The policy can then base its actions on the actuation envelope that is actually available instead of mistaking one temperature sensor for the entire plant state.
Practical responses are straightforward:
- Estimate unobserved rotor temperature with a winding sensor and a thermal observer.
- Apply continuous torque derating based on temperature and bus voltage.
- Vary $R$, $K_t$, current and voltage saturation, and thermal state together during training and simulation.
- Optimize trajectories with RMS current and regenerative capability, not just peak torque.
- Use springs, counterbalances, brakes, and mechanically favorable postures so that static loads are not held only by motor current.
Heat is slower than the control cycle, which is why it is not immediately visible. That also makes it more dangerous. Torque ripple appears on an oscilloscope right away, and backlash can be felt by reversing direction by hand, but thermal drift hides behind the first few minutes of a successful demo.
The real actuator specification is not its cold-state peak torque. It is how much trustworthy torque it can produce repeatedly, after reaching thermal equilibrium, in the given environment and duty cycle.
References
[1] maxon motor ag, “maxon Motor Data and Operating Ranges.” Manufacturer PDF
[2] Y. Xiao and A. Griffo, “PWM-Based Flux Linkage and Rotor Temperature Estimations for Permanent Magnet Synchronous Machines,” IEEE Transactions on Power Electronics, vol. 35, no. 6, pp. 6061–6069, 2020. doi:10.1109/TPEL.2019.2948578, accepted manuscript
[3] S. Seok et al., “Design Principles for Energy-Efficient Legged Locomotion and Implementation on the MIT Cheetah Robot,” IEEE/ASME Transactions on Mechatronics, vol. 20, no. 3, pp. 1117–1129, 2015. doi:10.1109/TMECH.2014.2339013, MIT Open Access
[4] P. M. Wensing, A. Wang, S. Seok, D. Otten, J. Lang, and S. Kim, “Proprioceptive Actuator Design in the MIT Cheetah: Impact Mitigation and High-Bandwidth Physical Interaction for Dynamic Legged Robots,” IEEE Transactions on Robotics, vol. 33, no. 3, pp. 509–522, 2017. doi:10.1109/TRO.2016.2640183, MIT manuscript
[5] B. Katz, J. Di Carlo, and S. Kim, “Mini Cheetah: A Platform for Pushing the Limits of Dynamic Quadruped Control,” IEEE International Conference on Robotics and Automation, pp. 6295–6301, 2019. doi:10.1109/ICRA.2019.8793865
[6] S. Yu et al., “Quasi-Direct Drive Actuation for a Lightweight Hip Exoskeleton With High Backdrivability and High Bandwidth,” IEEE/ASME Transactions on Mechatronics, vol. 25, no. 4, pp. 1794–1802, 2020. doi:10.1109/TMECH.2020.2995134, arXiv
[7] B. Katz, “A Low Cost Modular Actuator for Dynamic Robots,” M.S. thesis, Massachusetts Institute of Technology, 2018. MIT DSpace