Simon-Says Calibrated Rigs

• Simon-Says calibrated rigs are self-calibration systems that utilize prescribed motion sequences and known forward models to estimate extrinsic parameters without external artifacts.
• The approach employs model-based techniques combining sensor projection models and rigid kinematic priors in both camera–arm and Lighthouse realizations.
• These systems offer operational autonomy by enabling in-field recalibration through scripted motions, reducing reliance on manual measurements or conventional targets.

“Simon-Says calibrated rigs” (Editor’s term) denotes calibration systems in which the calibration signal is generated by prescribed motion rather than by external calibration artifacts or manual surveying. In the camera–arm setting, the rig is commanded to hold a pose and rotate a wrist joint, then to repeat that motion from additional poses, so that known arm kinematics and image observations of feature arcs can be used to estimate the camera pose relative to the arm base (McGuire et al., 2018). In the Lighthouse setting, a calibration board carrying photodiodes is moved through a short sequence of varied poses, and measured azimuth–elevation data are used to infer the relative 6-DoF poses of transmitter devices without tape-measure or protractor measurements (Jansen et al., 2022). The common principle is self-calibration through structured excitation: the rig “reveals” its geometry by obeying a constrained motion script.

1. Conceptual scope and defining characteristics

A Simon-Says calibrated rig replaces external reference structure with motion structure. The decisive feature is not the sensor modality but the availability of a known forward model together with a controllable or guidable sequence of motions. In the camera–arm case, the known model is the arm kinematic chain and the camera intrinsics; in the Lighthouse case, it is the sweep timing geometry together with the known 3D layout of the photodiodes on the calibration board. In both cases, the unknown is an extrinsic transform in $SE(3)$, and the observations are generated while the system executes or is guided through a constrained trajectory (McGuire et al., 2018, Jansen et al., 2022).

This class of systems is distinct from conventional target-based calibration. The camera–arm method is explicitly positioned as a way to recover the camera pose relative to the base of the arm’s kinematic chain “without relying on outside artifacts such as calibration targets,” and it uses “ambient features” on the end effector rather than precision fiducials (McGuire et al., 2018). The Lighthouse method addresses a different manual bottleneck: the position and orientation of transmitter devices had previously been measured by hand, a procedure described as time consuming, error-prone, and in some environments not possible (Jansen et al., 2022).

The phrase also implies a specific operational style. The rig is not expected to solve calibration from arbitrary passive observation; it is expected to perform, or to be led through, deliberate motions. In one realization, “Simon” is the robot controller commanding a final revolute joint while all other joints remain fixed. In the other, “Simon” is the software prompting an operator to move a board slowly through a tracking volume. This suggests that the term describes a methodological pattern rather than a single algorithmic family.

2. Shared geometric and estimation framework

The two realizations share a common decomposition: a known sensor-side projection model, a known rigid-body or kinematic prior, and an optimization that aligns predicted observations with measured ones. The camera–arm system seeks the transform $\mathbf{T}_{AC} \in SE(3)$, or equivalently a pose vector $\mathbf{x} = [x, y, z, \phi, \theta, \psi]$, between the arm base and the camera. Its arm model is expressed as

$T_{(i-1)(i)} = L_i T_{\theta_i},$

where $L_i$ is a fixed link transform and $T_{\theta_i}$ is the transform for the joint rotation angle $\theta_i$. The last joint is a wrist-type revolute joint with a rigidly mounted end effector, and points of the form $[0,0,z']^T$ lie on that joint’s rotation axis in the joint frame (McGuire et al., 2018).

The Lighthouse system formulates an analogous forward model for angular measurements. For a Lighthouse pose $\vec L = (x, y, z, \alpha, \beta, \gamma)$, a board pose $\vec P$, and known diode geometry, the per-diode measurement model is

$\mathbf{T}_{AC} \in SE(3)$0

with all $\mathbf{T}_{AC} \in SE(3)$1 diode pairs collected into

$\mathbf{T}_{AC} \in SE(3)$2

Pose estimation is then phrased as minimizing a Frobenius-norm discrepancy between predicted and measured angle matrices, first for board pose estimation and later for Lighthouse pose estimation over a sequence of board poses (Jansen et al., 2022).

In both systems, calibration is therefore neither purely geometric nor purely statistical. The geometry defines what motions are informative, while the estimator determines which extrinsic parameters best explain the accumulated observations. A plausible implication is that Simon-Says rigs are best understood as model-based self-calibration systems whose observability depends on the richness of the commanded trajectory.

3. Camera–arm realization through rotation identification

The camera–arm realization is organized around a highly constrained motion primitive: images are captured while the end effector rotates about the axis of the last joint and the remainder of the arm is motionless. That procedure is then repeated for $\mathbf{T}_{AC} \in SE(3)$3 distinct arm poses. For each pose, the arm kinematics provide the wrist axis in the world frame, while image tracks on the end effector provide visual evidence of the projected axis (McGuire et al., 2018).

The visual signal arises from conics. A fixed 3D point on the end effector rotates about the wrist axis, traces a 3D circle, and under perspective projection generates a 2D conic, generically an ellipse. The method tracks “ambient features” on the end effector using Lucas–Kanade optical flow after feature detection, and it treats each valid trajectory as an arc. The estimation pipeline is stated explicitly as: arc identification; circle estimation; estimation of the rotation axis by vision; rotation axis estimation using forward kinematics; and estimation of $\mathbf{T}_{AC} \in SE(3)$4 (McGuire et al., 2018).

For a single wrist-rotation measurement, the vision model uses a parameter vector $\mathbf{T}_{AC} \in SE(3)$5 with $\mathbf{T}_{AC} \in SE(3)$6 terms: an $\mathbf{T}_{AC} \in SE(3)$7 point on a 3D line, two orientation angles, and for each of $\mathbf{T}_{AC} \in SE(3)$8 feature tracks a radius and a displacement from the origin point. The essential structural prior is coaxiality: all candidate circles share a common 3D rotation axis. Candidate points on the circles are projected into the image plane according to the camera model, and ellipses are fit using Fitzgibbon’s direct least-squares method. The recovered axis projection is represented in 2D point-slope form as $\mathbf{T}_{AC} \in SE(3)$9, which becomes the vision-side observation of the wrist axis.

The final extrinsic estimate is obtained by aligning those image-line observations with the wrist axis predicted by forward kinematics under a candidate $\mathbf{x} = [x, y, z, \phi, \theta, \psi]$0. The residual is constructed from perpendicular distances between projected test points on the kinematic axis and the observed image line $\mathbf{x} = [x, y, z, \phi, \theta, \psi]$1: $\mathbf{x} = [x, y, z, \phi, \theta, \psi]$2 These distances are stacked into a residual vector $\mathbf{x} = [x, y, z, \phi, \theta, \psi]$3, and the cost is the weighted quadratic form $\mathbf{x} = [x, y, z, \phi, \theta, \psi]$4. Under Gaussian noise this is a maximum-likelihood estimator. Optimization uses Levenberg–Marquardt with random restarts, and the Jacobian is computed with the complex-step method. The covariance estimate is taken from the diagonal of $\mathbf{x} = [x, y, z, \phi, \theta, \psi]$5 (McGuire et al., 2018).

This realization exemplifies the Simon-Says pattern in a particularly literal form: the system does not require a checkerboard, fiducial, or motion-capture reference. The only calibration structure is the instruction sequence itself—hold still, rotate the last joint, move to another pose, and repeat.

4. Lighthouse realization through guided board trajectories

The Lighthouse realization addresses a low-cost 6-DoF tracking system built from HTC Vive “Lighthouse” transmitter devices and custom receiver boards. Each Lighthouse emits synchronization pulses and rotating infrared laser sweeps; the receiver measures the time difference $\mathbf{x} = [x, y, z, \phi, \theta, \psi]$6 between the sync pulse and the laser crossing with a microcontroller sampling at $\mathbf{x} = [x, y, z, \phi, \theta, \psi]$7, and converts that delay into azimuth and elevation-like angles (Jansen et al., 2022).

Calibration uses a dedicated board with $\mathbf{x} = [x, y, z, \phi, \theta, \psi]$8 photodiodes whose geometry is known in the board frame. During an approximately $\mathbf{x} = [x, y, z, \phi, \theta, \psi]$9 acquisition, the user moves this board freely while two Lighthouses operate in a master/slave configuration. Because each sweep produces either azimuth or elevation data and because master and slave sweeps alternate, the raw measurements are interleaved. The method reconstructs full angle observations by interpolation; the authors report that “full interpolation” was slightly more accurate than a half-density variant, although the differences were small. One full Lighthouse measurement cycle is about $T_{(i-1)(i)} = L_i T_{\theta_i},$0, the laser is visible between about $T_{(i-1)(i)} = L_i T_{\theta_i},$1 and $T_{(i-1)(i)} = L_i T_{\theta_i},$2 into the cycle, and the maximum time difference between consecutive usable measurements can be approximately $T_{(i-1)(i)} = L_i T_{\theta_i},$3. At $T_{(i-1)(i)} = L_i T_{\theta_i},$4, this corresponds to about $T_{(i-1)(i)} = L_i T_{\theta_i},$5 of board motion during the gap (Jansen et al., 2022).

The estimation then proceeds in stages. First, for each Lighthouse separately, the board pose $T_{(i-1)(i)} = L_i T_{\theta_i},$6 is estimated at each reconstructed time step by minimizing

$T_{(i-1)(i)} = L_i T_{\theta_i},$7

with the Lighthouse pose fixed at the origin of its own local frame. This produces two board trajectories, one in the master frame and one in the slave frame. Second, the trajectories are time-aligned by linear interpolation. Third, a weighted Kabsch alignment is applied to the board positions in the two trajectories, using the per-measurement pose estimation error $T_{(i-1)(i)} = L_i T_{\theta_i},$8 as weights, to obtain a best-fit rigid transform between the slave and master paths. That transform supplies an initial estimate $T_{(i-1)(i)} = L_i T_{\theta_i},$9 of the slave Lighthouse pose in the master frame. Finally, the slave pose is refined by minimizing

$L_i$0

over all measurements (Jansen et al., 2022).

The user-facing interpretation is explicitly compatible with Simon-Says-style prompting. The procedure can be framed as prompts such as “move it slowly through the air in front of both base stations,” “raise it above your head and lower it to waist level,” or “sweep it left to right and right to left.” The algorithm does not require one exact trajectory; it requires a rich set of poses. The reported experiments varied motion patterns, speeds, the distance to the Lighthouses between $L_i$1 and $L_i$2, and the Lighthouse separation between $L_i$3 and $L_i$4, and no correlation with final calibration accuracy was detected within that tested range (Jansen et al., 2022).

5. Observability, degeneracy, and practical constraints

The defining promise of Simon-Says calibration is target-free or survey-free operation, but neither realization is unconstrained. Both depend on non-degenerate excitation. In the camera–arm system, a specific pathological case is the “single crossing scenario,” in which the end effector maintains the same $L_i$5 position while changing only orientation. In that case, the location of the camera is underdetermined: camera orientation can be recovered, but position cannot. The paper states that a minimum of three measurements are required, with one measurement having a different position from the other two, in order to avoid this condition (McGuire et al., 2018).

The Lighthouse system exhibits an analogous dependence on trajectory richness. Although degenerate trajectories are not analyzed as extensively, the calibration logic depends on fitting a rigid transform between two sampled board trajectories. Motion confined to a small region, motion only along one straight line, or motion with little rotational variation leads to poor conditioning. Interpolation also imposes a speed-related limit: if the board moves very fast, the pose change between complementary azimuth and elevation measurements increases, which can degrade the reconstructed observations. The paper therefore recommends low movement speeds during calibration, even though no noticeable speed influence on precision was found in the reported experiments (Jansen et al., 2022).

A common misconception is that targetless calibration is therefore model-free calibration. The two systems show the opposite. The camera–arm method requires known arm kinematics, known joint angles, a rigid wrist and end effector, and known camera parameters with rectified image data. The Lighthouse method requires a rigid board with known diode positions, stable timing, and an assumed Lighthouse sweep geometry. This suggests that Simon-Says rigs transfer burden from external artifacts to internal modeling accuracy. Where those internal models are imperfect, the papers identify corresponding failure modes: correlated line-parameter errors and optimistic covariance estimates in the camera–arm method, and unmodeled Lighthouse intrinsic deviations in the Lighthouse method (McGuire et al., 2018, Jansen et al., 2022).

6. Accuracy, comparative position, and research significance

The two case studies report practically useful calibration performance under their respective assumptions. In the camera–arm simulation, six rotation observations were used and synthetic arcs were perturbed by zero-mean Gaussian noise with variances $L_i$6. Increasing tracking noise spread the fitted line-parameter errors and introduced correlations, but the reported variance of the expected errors remained within several centimeters even at the higher noise level. In the real-world experiment, conducted with an Asus Xtion Pro RGB-D camera and a Kinova Jaco2 6-DoF arm, the recovered pose was consistent with ruler-based estimates of $L_i$7; the reported covariance, on the order of $L_i$8, was explicitly described as excessively optimistic (McGuire et al., 2018).

The Lighthouse evaluation used a Qualisys motion-capture system as ground truth across $L_i$9 configurations, with $T_{\theta_i}$0 calibration runs per configuration. The overall mean absolute translation errors were $T_{\theta_i}$1 in $T_{\theta_i}$2, $T_{\theta_i}$3 in $T_{\theta_i}$4, and $T_{\theta_i}$5 in $T_{\theta_i}$6. The overall mean absolute rotation errors were $T_{\theta_i}$7 in $T_{\theta_i}$8, $T_{\theta_i}$9 in $\theta_i$0, and $\theta_i$1 in $\theta_i$2. Average standard deviations across setups were roughly $\theta_i$3, $\theta_i$4, and $\theta_i$5 for translation and about $\theta_i$6–$\theta_i$7 for rotation. Simulation produced lower MAEs, around $\theta_i$8–$\theta_i$9 and $[0,0,z']^T$0–$[0,0,z']^T$1, indicating that the simulation underestimated real-world effects. The authors state that the automatic procedure is more accurate than manual measurement and that some configurations could not be measured manually at all (Jansen et al., 2022).

Relative to traditional approaches, the camera–arm work occupies the hand–eye calibration literature but replaces target pose observations with rotation-axis observations derived from conics. The Lighthouse work extends beyond the usual “known base station pose, estimate object pose” formulation toward automatic recovery of transmitter poses from angle measurements and a moving board. In both cases, the principal significance lies in operational autonomy: calibration can be run outside a laboratory-style setup, after a rig is moved, or whenever drift or reconfiguration makes recalibration necessary. A plausible implication is that Simon-Says calibrated rigs are especially well suited to field robotics, room-scale tracking, and other environments in which calibration must be repeatable, low-overhead, and executable by the system itself or by a lightly guided operator.