[Robot Hardware 06] - Actuators (5): Backlash
How gear clearance, preload, and Harmonic Drive lost motion undermine precision and external-force estimation
If you move a robot arm slowly in one direction and then reverse it, the motor may move first while the link remains still for a short interval. The controller has already changed the position command and the motor encoder reports motion, but the joint does not receive the desired movement until the opposite tooth faces at the output shaft make contact.
This interval is commonly called backlash. In an actual actuator, however, the direction-reversal error also includes the geometric clearance of the tooth faces, elastic deformation of shafts, bearings, and gears, friction, assembly error, and hysteresis.[1,2] Calling all of these effects one dead zone makes it difficult to explain why the error of the same robot changes with load and time.
Why Leave a Gap Between Gear Teeth on Purpose?
Making two meshing gears exactly match their theoretical geometry without any gap may sound more precise, but real machines do not move that way. The reasons for requiring suitable clearance are practical:
- Every tooth profile has machining and pitch errors.
- Shafts and bearings have eccentricity, runout, and assembly tolerances.
- Teeth, shafts, and housings deform elastically under load.
- Different materials and components expand by different amounts as temperature changes.
- Space is needed for the lubricant film and for contamination to escape.
Some clearance is needed to ensure that the tooth faces do not jam or interfere even at worst-case tolerance. If it is too small, efficiency and life decrease, heat rises, and the reducer may bind at a particular position or show excessive resistance. If it is too large, lost motion and impact increase during direction reversal. Backlash is not simply a defect caused by machining cheaply; it is also design margin between manufacturability, lubrication, heat, and reliability.
Backlash, Compliance, and Hysteresis Are Not the Same Thing
Let the difference between input angle $\theta_m/N$ and output angle $\theta_j$ be transmission deflection:
\[\delta=\frac{\theta_m}{N}-\theta_j\]The simplest backlash model has almost no transmitted torque within a width $2b$ and develops stiffness $k$ after one of the two tooth faces is contacted:
\[\tau_t(\delta)= \begin{cases} k(\delta-b), & \delta>b \\ 0, & |\delta|\le b \\ k(\delta+b), & \delta<-b \end{cases}\]This is a memoryless dead-zone model in which $\tau_t$ is uniquely determined once $\delta$ is known. In an actual transmission, seal, bearing, and grease friction exists even before tooth contact, and teeth, shafts, and housings deform nonlinearly after contact. To represent a direction-dependent loop, the tooth-contact state must be included as an internal variable:
\[\tau_t=f(\delta,z),\qquad z\in\{-1,0,+1\}\]$z$ represents negative tooth-face contact, the free-clearance interval, and positive tooth-face contact. If the system does not retrace the same path when it reverses direction, a hysteresis loop appears in the torque–deflection curve.[1–3]
Distinguishing the terms makes diagnosis easier.
| Phenomenon | Meaning |
|---|---|
| backlash | Geometric clearance before the opposite tooth face makes contact |
| compliance | Elastic deformation caused by torque |
| friction | Internal resistance that obstructs the start and continuation of motion |
| hysteresis | Torque–displacement loop that depends on approach direction and previous load |
| lost motion | Total input–output position loss measured in a defined reversal test |
| kinematic error | Transmission error that repeats synchronously with rotational position |
Lost motion can include elastic deformation, friction history, and assembly deformation in addition to backlash. Therefore, “zero backlash” in a datasheet and lost motion measured during an actual torque reversal may not be the same number. Manufacturers and reducer types may also use different test definitions for backlash, hysteresis loss, and lost motion, so the measurement procedure and load condition must be checked together.
Does Preloading Both Tooth Faces with a Spring Solve the Problem?
A split gear or anti-backlash gear divides one gear into two halves and preloads the halves in opposite directions with a torsion spring. One half presses the leading tooth face and the other presses the trailing face, maintaining contact on both sides under no load.
This approach is useful where load is small and direction-reversal accuracy matters, such as in a positioning stage. It does not “delete” clearance so much as maintain two contacts with spring force.
- Higher preload increases tooth friction and bearing load.
- More friction and sliding worsen efficiency, temperature, and wear.
- Once external torque exceeds preload torque, one gear unloads and the mechanism switches to single-face contact; lost motion can reappear at a later load reversal.
- Spring fatigue and wear cause preload to change over time.
For joints that receive large bidirectional torque and impact, such as a humanoid knee or hip, “make the spring stronger” is not a simple solution. The tooth faces and bearings must continuously carry internal load just to maintain the preload.
Is Harmonic Drive Really Zero Backlash?
The strain-wave gear structure commonly called Harmonic Drive deforms the flexspline into an ellipse and meshes it with the circular spline in two regions at the same time. Unlike a conventional pair of spur gears, it can nearly eliminate the geometric clearance through which the gears move freely toward the opposite tooth face during reversal.
That does not mean the input and output have a perfectly rigid relationship. The thin flexspline and wave-generator bearing deform under load, and many contact surfaces and the lubricant create friction history. Experimental Harmonic Drive models include nonlinear stiffness, friction, hysteresis, and periodic transmission error.[3,4] Manufacturer data also treats zero backlash and torsional stiffness or hysteresis loss as separate characteristics.[5]
The following two statements can therefore both be true:
- The reducer has almost no tooth backlash in the traditional sense.
- Output angle after torque reversal still has lost motion and hysteresis.
From a robot-control perspective, the width of the load–displacement loop, load-dependent torsional stiffness, input/output encoder difference, and repeatable transmission error are more useful than marketing terms.
How Do a Few Arcminutes of One Joint Become Large at the End Effector?
For a small joint error $\delta q$, end-effector error is transmitted through the Jacobian to first order:
\[\delta x\approx J(q)\,\delta q\]The important point is that joint errors are not simply added as scalars. The Cartesian effect of the same $0.1^\circ$ differs between an extended arm and a folded arm, and the projection into translation and orientation depends on the direction of each joint axis. A rough upper bound on position error from revolute joints grows with the effective lengths of the downstream links.
\[\lVert\delta p\rVert \lesssim \sum_i l_{i,\mathrm{eff}}\,|\delta q_i|\]A six-axis industrial arm has six joints, but a humanoid contact chain can be longer. From a foot fixed to the ground through ankle, knee, hip, pelvis, torso, shoulder, elbow, and wrist to the hand, the direction history of multiple joints affects the hand’s position and force at once.
In a floating-base system such as a humanoid, Cartesian error propagation must include not only joint error but also base-pose estimation error and which contact points are actually constrained.
This is also where repeatability and absolute accuracy diverge. If the robot approaches the same pose from the same direction every time, the same tooth faces can contact and the result may repeat well. Approaching from the opposite direction, or changing payload, gravity direction, or contact force, uses different tooth faces and different elastic deformation. Good repeatability does not guarantee accuracy in a bidirectional force task.
From the Control Perspective: When Motor and Output Shaft Separate
Backlash makes control difficult for more than the additional position error. At the instant of direction reversal, transmitted torque between the motor side and output shaft can temporarily disappear, allowing the two inertias to move independently.
Define motor position referred to the output shaft as
\[q_m=\frac{\theta_m}{N}\]The difference between motor-side position and actual output-joint position is transmission deflection:
\[\delta=q_m-q_j\]Let output-referred motor-rotor inertia be $J_m^*=N^2J_m$, and let $\tau_{\mathrm{act}}$ be drive torque referred to the output shaft. A simplified motor-side dynamic equation is
\[J_m^*\ddot q_m=\tau_{\mathrm{act}}-\tau_t\]The output-link dynamics can be written as
\[J_j\ddot q_j+B_j\dot q_j+\tau_{\mathrm{load}}=\tau_t+\tau_{\mathrm{ext}}\]Here, $\tau_t$ is the torque actually transmitted through the reducer. When the teeth are fully engaged, torque passes between motor and output shaft according to transmission stiffness. Inside the backlash gap, the following state can occur:
\[|\delta|\le b \quad\Rightarrow\quad \tau_t\approx0\]In this interval, the motor can move and current can flow without the same motion and torque reaching the output link. A position controller using only a motor-side encoder may see motor position error decrease quickly and conclude that tracking is normal, while the actual output joint has not moved yet.
When reversal continues and the opposite tooth face makes contact, transmitted torque forms again:
\[|\delta|>b \quad\Rightarrow\quad \tau_t\approx k\left(\delta-\operatorname{sgn}(\delta)b\right)\]If relative velocity accumulated by the motor and internal gears in the clearance interval remains, an impact torque can occur at tooth contact. The higher the position gain used to cross backlash quickly, the larger the relative speed and impact at contact can become. Repeated reversals can appear as a limit cycle or noise.
Within the backlash interval, one motor-side encoder position can correspond to multiple output-joint positions:
\[q_j\in[q_m-b,\ q_m+b]\]Motor-side position measurement alone therefore cannot uniquely determine the instantaneous output-joint position or the tooth face currently in contact. Dynamic history, inputs, contact models, and sufficient excitation can make the state observable again, but a single instantaneous measurement is missing the internal state that represents previous motion direction and torque history.
With a dual-encoder structure, both motor-side and output-side positions are measured and the following value can be calculated directly:
\[\delta=\frac{\theta_m}{N}-\theta_j\]This makes it possible to estimate whether the output shaft is within the clearance, which tooth face is in contact, and how much the transmission has deformed under load. But because backlash, elastic deformation, and friction coexist, calculating transmitted torque from $\delta$ alone still requires load history and a nonlinear transmission model.
Backlash is therefore not a static parameter that adds a fixed error to a position command. It is an internal state that changes the dynamic connection between motor and load, changing the meaning of control input and sensor measurement at the moment of direction reversal.
Backlash Breaks Observability for External-Force Sensing
A robot without a separate joint-torque sensor may estimate external force from motor current. Conceptually,
\[\hat{\tau}_{\mathrm{act}}\approx\hat{\eta}N\hat K_t i_q\] \[\hat{\tau}_{\mathrm{ext}} =\hat{\tau}_{\mathrm{act}} -\hat M(q)\ddot q -\hat C(q,\dot q)\dot q -\hat G(q) -\hat{\tau}_f\]Here, $\hat\eta$ is estimated transmission efficiency. After subtracting robot dynamics and friction, a filter or observer estimates the disturbance torque. The estimated joint torque can be converted into a Cartesian wrench using the Jacobian. Even with a well-calibrated rigid transmission, this is difficult because inertia, friction, and motor torque-constant error must be removed.[6]
The backlash interval creates a more fundamental problem:
\[\tau_m\ne0 \qquad\text{but}\qquad \tau_t\approx0\]Motor torque may be used only to accelerate the motor and internal gear inertia while not yet reaching the output shaft. Two additional observation failures occur.
First, while the motor crosses the clearance, current flows and the motor encoder moves, but the same motion or torque is not transmitted to the output shaft. Acceleration and current spikes at tooth impact can be mistaken for external contact.
Second, an external force can move the output link within the clearance while the motor shaft barely moves. A motor-side encoder then misses both the actual joint-angle change and external contact. In the backlash interval, motor-side position measurement alone cannot uniquely determine the instantaneous output-joint position or contact-tooth state.
A joint-torque sensor or series elastic element can help separate external force from internal friction. Adding one sensor does not eliminate the need for a model, however. The physical layer that the sensor sees depends on whether it is placed before or after the reducer.
Observed Lost Motion Changes with Time and Operating Conditions
There are several reasons not to treat backlash and a lost-motion curve measured on a new reducer as fixed parameters:
- Wear in bearings and gear teeth can increase clearance.
- Grease temperature and distribution can change friction and the hysteresis loop even at the same clearance.
- Preload springs and bearing preload can relax.
- Housing temperature and external force can change shaft alignment.
- Shock loads can leave local damage or permanent deformation.
Tooth wear and thermal expansion can change the actual clearance, while lubrication state, temperature, and preload can change the reversal loop at that same clearance. The real mapping is therefore closer to $\tau=f(\delta,z_{gear},T,t_{life})$, which includes current tooth contact, approach direction, load history, temperature, and operating time, than to $\tau=f(\delta)$. This is why backlash is modeled in the literature as a dynamic nonlinear effect with a hidden state.[1,2,7]
What Should Remain in the Design and Policy?
Mechanical design should measure load-dependent lost-motion curves, torsional stiffness, reversal life, and temperature range rather than one backlash number. In control, increasing gain indiscriminately near direction reversal can increase tooth collision and limit cycles. For observation, a motor/output dual encoder, torque sensor, or at least a direction-history state is preferable.
Simulation also needs more than random noise added to joint angle. At minimum, distinguish:
- clearance in which the output does not move until the opposite tooth face is contacted
- compliance that deforms according to load after contact
- hysteresis that differs with approach direction
- speed- and temperature-dependent friction
- parameter distributions that change with operating time
The central issue with backlash is not merely that “the angle is slightly wrong.” It is that there is an interval in which the output-link state and external force cannot be uniquely determined from the motor-side state alone. In a long kinematic chain and contact controller, this small unobservable interval becomes end-effector position error, force-estimation error, and impact.
Next post: [Robot Hardware 07] - Actuators (6): Heat
References
[1] M. Nordin and P.-O. Gutman, “Controlling Mechanical Systems With Backlash—A Survey,” Automatica, vol. 38, no. 10, pp. 1633–1649, 2002. doi:10.1016/S0005-1098(02)00047-X
[2] M. Ruderman, F. Hoffmann, and T. Bertram, “Modeling and Identification of Elastic Robot Joints With Hysteresis and Backlash,” IEEE Transactions on Industrial Electronics, vol. 56, no. 10, pp. 3840–3847, 2009. doi:10.1109/TIE.2009.2015752
[3] R. Dhaouadi, F. H. Ghorbel, and P. S. Gandhi, “A New Dynamic Model of Hysteresis in Harmonic Drives,” IEEE Transactions on Industrial Electronics, vol. 50, no. 6, pp. 1165–1171, 2003. doi:10.1109/TIE.2003.819661
[4] C. Preissner, T. J. Royston, and D. Shu, “A High-Fidelity Harmonic Drive Model,” Journal of Dynamic Systems, Measurement, and Control, vol. 134, no. 1, 011002, 2012. doi:10.1115/1.4005041
[5] Harmonic Drive LLC, “Reducer Catalog: Component Sets FB — Engineering Data.” Manufacturer PDF
[6] A. Wahrburg, J. Bös, K. D. Listmann, F. Dai, B. Matthias, and H. Ding, “Motor-Current-Based Estimation of Cartesian Contact Forces and Torques for Robotic Manipulators and Its Application to Force Control,” IEEE Transactions on Automation Science and Engineering, vol. 15, no. 2, pp. 879–886, 2018. doi:10.1109/TASE.2017.2691136
[7] E. Giovannitti, S. Nabavi, G. Squillero, and A. Tonda, “A Virtual Sensor for Backlash in Robotic Manipulators,” Journal of Intelligent Manufacturing, vol. 33, no. 7, pp. 1921–1937, 2022. doi:10.1007/s10845-022-01934-z