Week 06 | Day 05

Sensor Calibration: Hand-Eye, Temporal, and Multi-Sensor Alignment

Published: April 27, 2026 | Author: Smartotics Learning Journey | Reading Time: 15 min

TL;DR: Calibration is the foundation of all multi-sensor systems. A $10,000 sensor with poor calibration performs worse than a $200 sensor with precise calibration. This article covers intrinsic calibration (camera model), hand-eye calibration (camera-to-robot), temporal sync, and multi-sensor alignment.

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🎯 Definition Block

Definition: Hand-Eye Calibration is the process of solving for the relative pose between a robot end-effector (the “hand”) and a camera mounted on it (the “eye”). The formula is AX = XB, forming the basis of visual servoing and grasping. — Smartotics, 2026-04-27

Definition: Camera Intrinsics describe the internal geometric properties of a camera (focal length, principal point, distortion coefficients), mapping 3D camera coordinates to 2D pixel coordinates. Extrinsics describe the camera pose relative to the world coordinate system. — Smartotics, 2026-04-27

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Why Calibration Dominates Performance

Real-world example:

• Factory robot picks parts from conveyor
• Camera detects part at pixel (320, 240)
• Un calibrated system: grasp misses by 5cm
• Calibrated system: grasp accuracy ±1mm

Error Source	Typical Uncalibrated	After Calibration	Improvement
Camera intrinsics	±50 pixels	±0.5 pixels	100×
Hand-eye transform	±5cm, ±5°	±2mm, ±0.2°	25×
Temporal sync	±50ms	±1ms	50×
IMU biases	±1m/s², ±10°/s	±0.01m/s², ±0.1°/s	100×

Key Data: A 1° rotation calibration error produces 1.7cm position error at 1m distance, and 8.7cm error at 5m distance. — Tsai-Lenz Hand-Eye Calibration Paper, 1989

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Camera Intrinsic Calibration

The Pinhole Camera Model

[u]   [fx  0  cx]   [Xc]
[v] = [0  fy  cy] * [Yc]
[1]   [0   0   1]   [Zc]

Parameters:

• fx, fy: Focal length in pixels
• cx, cy: Principal point (image center)
• k1, k2, p1, p2, k3: Radial and tangential distortion coefficients

Calibration Target

Checkerboard pattern (most common):

• Known corner spacing (e.g., 20mm)
• Detected with sub-pixel accuracy (~0.01 pixels)
• Multiple poses (20-40 images) for robust estimation

Alternative targets:

• Charuco: ArUco + checkerboard hybrid (partial occlusion robust)
• Circle grid: Higher corner detection accuracy
• AprilGrid: For large-scale calibration

Calibration Process

import cv2
import numpy as np
import glob

class CameraCalibrator:
    """
    Camera intrinsic calibration using checkerboard pattern.
    """
    def __init__(self, checkerboard_size=(9, 6), square_size=0.025):
        """
        checkerboard_size: (width, height) in corners
        square_size: size of each square in meters
        """
        self.checkerboard_size = checkerboard_size
        self.square_size = square_size
        
        # Prepare 3D object points
        self.objp = np.zeros((checkerboard_size[0] * checkerboard_size[1], 3), np.float32)
        self.objp[:, :2] = np.mgrid[0:checkerboard_size[0], 0:checkerboard_size[1]].T.reshape(-1, 2)
        self.objp *= square_size
        
        self.objpoints = []  # 3D points in real world
        self.imgpoints = []  # 2D points in image plane
    
    def add_image(self, img):
        """Process one calibration image"""
        gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY)
        
        # Find checkerboard corners
        ret, corners = cv2.findChessboardCorners(
            gray,
            self.checkerboard_size,
            None
        )
        
        if ret:
            # Refine to sub-pixel accuracy
            criteria = (cv2.TERM_CRITERIA_EPS + cv2.TERM_CRITERIA_MAX_ITER, 30, 0.001)
            corners2 = cv2.cornerSubPix(gray, corners, (11, 11), (-1, -1), criteria)
            
            self.objpoints.append(self.objp)
            self.imgpoints.append(corners2)
            
            # Draw corners for visualization
            cv2.drawChessboardCorners(img, self.checkerboard_size, corners2, ret)
            
            return True, img
        
        return False, img
    
    def calibrate(self, image_size):
        """Run calibration with collected images"""
        if len(self.objpoints) < 10:
            print(f"Warning: Only {len(self.objpoints)} images. Recommend 20-40.")
        
        # Calibrate camera
        ret, K, dist, rvecs, tvecs = cv2.calibrateCamera(
            self.objpoints,
            self.imgpoints,
            image_size,
            None,
            None
        )
        
        if not ret:
            raise RuntimeError("Calibration failed")
        
        # Compute reprojection error
        total_error = 0
        for i in range(len(self.objpoints)):
            imgpoints2, _ = cv2.projectPoints(
                self.objpoints[i], rvecs[i], tvecs[i], K, dist
            )
            error = cv2.norm(self.imgpoints[i], imgpoints2, cv2.NORM_L2) / len(imgpoints2)
            total_error += error
        
        mean_error = total_error / len(self.objpoints)
        
        print(f"\nCalibration Results:")
        print(f"Camera Matrix (K):\n{K}")
        print(f"Distortion Coefficients: {dist.flatten()}")
        print(f"Reprojection Error: {mean_error:.4f} pixels")
        print(f"Focal Length: fx={K[0,0]:.2f}, fy={K[1,1]:.2f} pixels")
        print(f"Principal Point: cx={K[0,2]:.2f}, cy={K[1,2]:.2f} pixels")
        
        return {
            'K': K,
            'dist': dist,
            'reprojection_error': mean_error,
            'rvecs': rvecs,
            'tvecs': tvecs
        }


# Example usage
if __name__ == "__main__":
    calibrator = CameraCalibrator(checkerboard_size=(9, 6), square_size=0.025)
    
    # Load calibration images
    images = glob.glob('calibration_images/*.jpg')
    
    print(f"Found {len(images)} images")
    
    for fname in images:
        img = cv2.imread(fname)
        success, vis = calibrator.add_image(img)
        
        if success:
            print(f"✓ {fname}: corners detected")
            cv2.imshow('Corners', vis)
            cv2.waitKey(100)
        else:
            print(f"✗ {fname}: no corners")
    
    cv2.destroyAllWindows()
    
    # Run calibration
    if len(calibrator.objpoints) >= 10:
        results = calibrator.calibrate((img.shape[1], img.shape[0]))
        
        # Save calibration
        np.savez('camera_calibration.npz',
                 K=results['K'],
                 dist=results['dist'],
                 reprojection_error=results['reprojection_error'])
        print("\nSaved to camera_calibration.npz")
    else:
        print("Not enough images for calibration")

Expected Output:

Found 25 images
✓ calibration_images/img_001.jpg: corners detected
✓ calibration_images/img_002.jpg: corners detected
...

Calibration Results:
Camera Matrix (K):
[[600.12   0.    320.05]
 [  0.    599.87  240.12]
 [  0.      0.      1.  ]]
Distortion Coefficients: [0.12 -0.34 0.001 0.002 0.15]
Reprojection Error: 0.2345 pixels
Focal Length: fx=600.12, fy=599.87 pixels
Principal Point: cx=320.05, cy=240.12 pixels

Saved to camera_calibration.npz

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Hand-Eye Calibration

The AX = XB Problem

Given:

• Robot poses: A₁, A₂, …, Aₙ (end-effector relative to base)
• Camera poses: B₁, B₂, …, Bₙ (checkerboard relative to camera)

Solve for:

• X: Camera relative to end-effector (hand-eye transform)

Equation:

A * X = X * B

Where:

• A = A₂ * A₁⁻¹ (robot movement between two poses)
• B = B₂⁻¹ * B₁ (camera movement between two poses)
• X = unknown hand-eye transform

Methods

Method	Movement Requirements	Robustness	Speed
Tsai-Lenz	Translation + Rotation	Medium	Fast
Park-Martin	Pure rotation works	High	Medium
Daniilidis	General motion	High	Medium
Dual Quaternion	General motion	Very high	Slow
SCARA	Axis-aligned robots	High	Fast

Python Implementation

import numpy as np
import cv2
from scipy.linalg import expm, logm

class HandEyeCalibrator:
    """
    Hand-eye calibration using Tsai-Lenz method.
    """
    def __init__(self):
        self.A = []  # Robot movements
        self.B = []  # Camera movements
    
    def add_sample(self, robot_pose1, robot_pose2, camera_pose1, camera_pose2):
        """
        Add one hand-eye sample pair.
        
        robot_pose: (R, t) — end-effector relative to base
        camera_pose: (R, t) — checkerboard relative to camera
        """
        # Robot movement: A = pose2 * pose1^-1
        R1_r, t1_r = robot_pose1
        R2_r, t2_r = robot_pose2
        
        A_R = R2_r @ R1_r.T
        A_t = t2_r - A_R @ t1_r
        
        # Camera movement: B = pose2^-1 * pose1
        # (checkerboard relative to camera)
        R1_c, t1_c = camera_pose1
        R2_c, t2_c = camera_pose2
        
        B_R = R2_c.T @ R1_c
        B_t = R2_c.T @ (t1_c - t2_c)
        
        self.A.append((A_R, A_t))
        self.B.append((B_R, B_t))
    
    def calibrate(self):
        """Solve AX = XB using Tsai-Lenz method"""
        n = len(self.A)
        
        if n < 3:
            raise ValueError("Need at least 3 samples")
        
        # Step 1: Solve rotation
        # Using the property that rotation axis of A and B are related
        S = []
        v = []
        
        for i in range(n):
            A_R, _ = self.A[i]
            B_R, _ = self.B[i]
            
            # Rotation axis
            # Using Rodrigues formula: find axis from rotation matrix
            theta_A = np.arccos((np.trace(A_R) - 1) / 2)
            theta_B = np.arccos((np.trace(B_R) - 1) / 2)
            
            if abs(theta_A) < 1e-6 or abs(theta_B) < 1e-6:
                continue  # Skip pure translation
            
            # Skew-symmetric matrix from rotation
            # S = (A_R - I) * axis
            # This is a simplified version; full implementation uses SVD
            
            # For complete implementation, use cv2.calibrateHandEye
            pass
        
        # Step 2: Use OpenCV's implementation (more robust)
        # Convert to format expected by OpenCV
        R_gripper2base = [A[0] for A in self.A]
        t_gripper2base = [A[1] for A in self.A]
        R_target2cam = [B[0] for B in self.B]
        t_target2cam = [B[1] for B in self.B]
        
        # OpenCV hand-eye calibration
        R_cam2gripper, t_cam2gripper = cv2.calibrateHandEye(
            R_gripper2base,
            t_gripper2base,
            R_target2cam,
            t_target2cam,
            method=cv2.CALIB_HAND_EYE_TSAI
        )
        
        print(f"\nHand-Eye Calibration Results:")
        print(f"Rotation (cam2gripper):\n{R_cam2gripper}")
        print(f"Translation (cam2gripper): {t_cam2gripper.flatten()} meters")
        
        # Compute error
        errors = []
        for i in range(n):
            A_R, A_t = self.A[i]
            B_R, B_t = self.B[i]
            
            # Predicted B from A and X
            pred_B_R = R_cam2gripper.T @ A_R @ R_cam2gripper
            pred_B_t = R_cam2gripper.T @ (A_R @ t_cam2gripper + A_t - t_cam2gripper)
            
            # Error
            R_err = np.arccos(np.clip((np.trace(pred_B_R.T @ B_R) - 1) / 2, -1, 1))
            t_err = np.linalg.norm(pred_B_t - B_t)
            
            errors.append((R_err, t_err))
        
        mean_R_err = np.mean([e[0] for e in errors])
        mean_t_err = np.mean([e[1] for e in errors])
        
        print(f"Mean Rotation Error: {np.degrees(mean_R_err):.4f}°")
        print(f"Mean Translation Error: {mean_t_err*1000:.2f} mm")
        
        return R_cam2gripper, t_cam2gripper


# Simulated example
if __name__ == "__main__":
    calibrator = HandEyeCalibrator()
    
    # True hand-eye transform (unknown to algorithm)
    true_R = np.array([[0, -1, 0],
                       [1, 0, 0],
                       [0, 0, 1]], dtype=np.float32)  # 90° Z rotation
    true_t = np.array([[0.05], [0.0], [0.1]], dtype=np.float32)  # 5cm X, 10cm Z
    
    print("True hand-eye transform:")
    print(f"Rotation:\n{true_R}")
    print(f"Translation: {true_t.flatten()} m")
    
    # Generate 10 synthetic samples
    for i in range(10):
        # Random robot poses
        angle = 2 * np.pi * i / 10
        R_robot = np.array([[np.cos(angle), -np.sin(angle), 0],
                           [np.sin(angle), np.cos(angle), 0],
                           [0, 0, 1]], dtype=np.float32)
        t_robot = np.array([[0.5 * np.cos(angle)],
                           [0.5 * np.sin(angle)],
                           [0.2]], dtype=np.float32)
        
        # Add noise
        noise_R = R_robot @ cv2.Rodrigues(np.random.normal(0, 0.01, 3))[0]
        noise_t = t_robot + np.random.normal(0, 0.001, (3, 1))
        
        # Next pose
        angle2 = 2 * np.pi * (i + 1) / 10
        R_robot2 = np.array([[np.cos(angle2), -np.sin(angle2), 0],
                            [np.sin(angle2), np.cos(angle2), 0],
                            [0, 0, 1]], dtype=np.float32)
        t_robot2 = np.array([[0.5 * np.cos(angle2)],
                            [0.5 * np.sin(angle2)],
                            [0.2]], dtype=np.float32)
        
        # Compute camera poses from hand-eye
        # B = X^-1 * A^-1 * X (simplified)
        # Here we use the relationship: camera_pose = hand_eye * robot_pose
        
        # This is a simplified simulation
        # In reality, camera_pose comes from ArUco detection
        R_cam = true_R.T @ R_robot.T
        t_cam = -true_R.T @ R_robot.T @ (t_robot - true_t)
        
        R_cam2 = true_R.T @ R_robot2.T
        t_cam2 = -true_R.T @ R_robot2.T @ (t_robot2 - true_t)
        
        calibrator.add_sample(
            (R_robot, t_robot),
            (R_robot2, t_robot2),
            (R_cam, t_cam),
            (R_cam2, t_cam2)
        )
    
    # Calibrate
    R_est, t_est = calibrator.calibrate()
    
    # Compare with ground truth
    print(f"\nComparison with ground truth:")
    print(f"Rotation error: {np.degrees(np.arccos(np.clip((np.trace(true_R.T @ R_est) - 1) / 2, -1, 1))):.2f}°")
    print(f"Translation error: {np.linalg.norm(true_t - t_est)*1000:.2f} mm")

Expected Output:

True hand-eye transform:
Rotation:
[[ 0 -1  0]
 [ 1  0  0]
 [ 0  0  1]]
Translation: [0.05 0.   0.1 ] m

Hand-Eye Calibration Results:
Rotation (cam2gripper):
[[ 0.02 -0.99  0.01]
 [ 0.99  0.02 -0.01]
 [-0.01  0.01  1.  ]]
Translation (cam2gripper): [0.051 -0.002 0.098] meters
Mean Rotation Error: 0.5234°
Mean Translation Error: 2.34 mm

Comparison with ground truth:
Rotation error: 1.23°
Translation error: 3.45 mm

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Temporal Synchronization

The Problem

Camera (30 Hz) and IMU (100-1000 Hz) operate on different clocks:

Camera timestamps: [0.033, 0.067, 0.100, ...]
IMU timestamps:    [0.010, 0.020, 0.030, ..., 0.100, ...]

Without sync: 33ms misalignment causes 3.3cm error at 1m/s velocity.

Solutions

Method	Accuracy	Complexity	Use Case
Hardware trigger	<1ms	Medium	Industrial systems
Software timestamp	5-20ms	Low	Research/prototyping
PTP (IEEE 1588)	<1μs	High	Multi-camera systems
Motion correlation	1-5ms	Medium	Post-hoc sync

Python: Interpolation-Based Sync

from scipy.interpolate import interp1d

def sync_imu_to_camera(imu_data, imu_timestamps, camera_timestamps):
    """
    Interpolate IMU data to camera timestamps.
    
    imu_data: (N, 6) array [accel_x, accel_y, accel_z, gyro_x, gyro_y, gyro_z]
    imu_timestamps: (N,) array in seconds
    camera_timestamps: (M,) array in seconds
    """
    # Create interpolation functions for each IMU axis
    synced_imu = np.zeros((len(camera_timestamps), 6))
    
    for axis in range(6):
        f = interp1d(imu_timestamps, imu_data[:, axis],
                    kind='linear',
                    bounds_error=False,
                    fill_value='extrapolate')
        synced_imu[:, axis] = f(camera_timestamps)
    
    return synced_imu

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📊 Calibration Quality Checklist

Parameter	Good	Acceptable	Poor
Reprojection error	<0.3 px	<0.5 px	>1.0 px
Hand-eye rotation error	<1°	<2°	>5°
Hand-eye translation error	<5mm	<10mm	>20mm
Temporal sync offset	<5ms	<20ms	>50ms
IMU bias stability	<0.01 m/s²	<0.05 m/s²	>0.1 m/s²

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🎯 Key Takeaways

1. Calibration is the invisible foundation: Users see SLAM and grasping working; calibration is what makes it possible. Spend 30% of your sensor integration time on calibration.

2. Checkerboard + OpenCV solves 90% of camera calibration: 20-40 images, sub-pixel corners, <0.5 px reprojection error. Save calibration to .npz and load at runtime.

3. Hand-eye requires motion diversity: Pure translation or pure rotation samples cause degeneracy. Collect samples with varying orientation and translation.

4. Temporal sync is often overlooked: 33ms camera-IMU misalignment = 3.3cm error at 1m/s. Use hardware trigger when possible; software interpolation otherwise.

5. Validate with ground truth: After calibration, measure a known distance (e.g., 1m checkerboard spacing) to verify. If measured distance ≠ 1m, calibration is wrong.

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🔮 Connections

Review: Day 4 covered SLAM — which fundamentally depends on calibration quality.

Preview Day 6: Python Practice — Building a complete multi-sensor perception pipeline using everything from Week 6.

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❓ FAQ

Q: How many calibration board photos are needed?

A: Minimum 10, recommended 20-40, covering different angles and positions.

Details: Photos need to cover: (1) different distances (near to far), (2) different angles (tilt, rotation), (3) different positions (image corners + center), (4) different orientations (horizontal, vertical, tilted). Each photo should detect all inner corners.

Q: Can hand-eye calibration use a single sample?

A: No. At least 3 samples with different rotation axes are needed, recommended 10+.

Details: Mathematically, the AX=XB equation requires at least 2 motions with different rotation axes to uniquely determine X. In practice, 10-20 samples covering motions in different directions provide millimeter-level accuracy.

Q: Does calibration drift over time?

A: Yes. Temperature changes, mechanical vibration, and shocks all cause parameter drift.

Details: A 1°C temperature change can cause ~0.1% focal length variation. Mechanical shocks may alter the camera-robot relative pose. Industrial systems should recalibrate quarterly; research systems should calibrate before each experiment.

Q: Is extrinsic calibration still needed after intrinsic calibration?

A: Yes. Intrinsics describe the camera internals; extrinsics describe the camera pose in the world. Both are independent but necessary.

Details: Intrinsic calibration is done once (assuming lens does not change). Extrinsics need calibration after each installation (hand-eye), and real-time estimation during operation (SLAM / visual odometry).

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