## 4 Visual Odometry  
### Motion Model  
In contrast to a UAV mounted or hand held camera, the pose of a camera that is equipped on a forklift corresponds to its kinematic constraints. Therefore, the motion of the camera can be estimated using the motion model of the forklift [34]. First, the kinematic model of the forklift is analyzed as described below. The principle of the forklift robot with rear wheel steering is shown in Figure 3. \(O-X-Y\) and \(o-x-y\) represent the world coordinate system and the forklift coordinate system, respectively. The angle between the rear wheel of the forklift and the \(y\)-axis of the forklift coordinate system is denoted by \(\alpha\), and the counter-clockwise direction is positive. The velocity of the front drive shaft center of the forklift is defined as forklift speed \(v\), and the angle between the longitudinal axis of the forklift coordinate system \(y\) and the world coordinate system \(Y\) is defined as the course angle of the forklift, which is denoted by \(\theta\), taking the counter-clockwise direction as positive. The distance between the front and rear wheels of the forklift is defined as \(L\). It was assumed that the wheel will not slip, and the instantaneous rotation center of the forklift was the intersection point of the normal direction  
Figure 3: Schema of the forklift with a spin-turn steering mechanism. of the front and rear wheels. The movement of the forklift can easily be obtained in accordance with the following constraints:  
\[ \begin{bmatrix}\dot{x}\\ \dot{y}\\ \dot{\theta}\end{bmatrix} = \begin{bmatrix}v\cos\theta\\ v\sin\theta\\ \frac{\ddot{v}}{L}\tan\alpha\end{bmatrix}  \tag{9}\]  
This is the kinematic model of the wheeled vehicle, and it describes the relationship between the acceleration and motion of the vehicle. Because the camera and forklift were fixed to each other, the camera and the forklift could be regarded as a rigid body, so the motion of the camera accorded with the above kinematic constraints. The camera was mounted at the top center of lift, and its optical center coincided with the X-axis of the forklift. The distance between the lift and the Y-axis of the forklift was measured by a laser ruler, so the rigid transformation between them could be obtained. \({}^{V}\mathbf{T}_{C}\) represents the camera's pose in forklift coordinate, \({}^{W}\mathbf{T}_{V}\) represents the forklift's pose in the world coordinate, so the camera's pose in world \({}^{W}\mathbf{T}_{C}={}^{W}\mathbf{T}_{V}{}^{V}\mathbf{T}_{C}\). Therefore, the motion model of the forklift could be used to roughly estimate the pose of the camera. The result of pose estimation could be used as the initial value of the direct method to solve the pose of the camera [32] to improve the speed of solving the pose of the camera, to reduce the time of the nonlinear optimization algorithm, and to reduce the probability that the direct method optimization falls into the local optimum. In general, the acceleration of a vehicle is relatively stable in the course of its operation and does not change dramatically as in the case of a hand held camera or a camera deployed on a UAV. The steering angle \(\alpha\) and speed \(v\) of the forklift truck cannot change instantaneously. The derivatives of these variables are the control signals of the vehicle. \[\dot{\alpha}=u_{\alpha} \tag{10}\] \[\dot{v}=u_{v}\]  
This study assumed that the change rate of the forklift control signal remained unchanged within a short period; that is, the change in the control quantity in a certain period of time during the movement of the vehicle was linear. \[\dot{u}_{a_{i+1}}=\dot{u}_{a_{i}}=\delta u_{\alpha} \tag{11}\] \[\dot{u}_{v_{i+1}}=\dot{u}_{v_{i}}=\delta u_{v}\]  
The forklift's pose at \(t_{i}\) is \(\mathbf{T}_{i}\). The coordinates \(x_{i}\) and \(y_{i}\) and the course angle \(\theta_{i}\) of the forklift at \(t_{i}\) can be solved from \(\mathbf{T}_{i}\). The linear velocity of the vehicle is \(v_{i}\), and the angle of the wheel is \(\alpha_{i}\) at the time \(t_{i}\).