## Appendix B  
The following lemmas are proposed in this paper for the coupling process on the velocity layer. **Lemma A1**. : _For the IMU-camera measurement system in Figure 1, let \(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\) be the equivalent rotation of \(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\) described in the IMU frame. Assuming that \(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\), \(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\) and \(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\) are unit quaternions with \(\overline{\mathbf{q}}_{w}^{{}^{\prime}}=\overline{\mathbf{q}}_{w}^{{}^{\prime}}= \overline{\mathbf{q}}_{w}^{{}^{\prime}}\), \(\overline{\mathbf{q}}_{w}^{{}^{\prime}}=\left(\overline{\mathbf{q}}_{w}^{{}^{\prime}} \right)^{\mu_{\omega}}\otimes\left(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\right)^ {1-\mu_{\omega}}\), where \(\mu_{\omega}\in[0,1]\)._  
Proof of Lemma A1. : Rewriting the quaternions into axis-angle form [32] yields \(\overline{\mathbf{q}}_{w}^{{}^{\prime}}=\left[\cos\theta_{ic},\mathbf{r}_{ic}\sin \theta_{ic}\right]\), \(\overline{\mathbf{q}}_{w}^{{}^{\prime}}=\left[\cos\theta_{c},\mathbf{r}_{c}\sin \theta_{c}\right]\), and \(\mathbf{q}_{w}^{{}^{\prime}}=\left[\cos\theta_{i},\mathbf{r}_{i}\sin\theta_{i}\right]\), with \(\theta_{c}=\theta_{i}=\theta_{ic}=\theta\), and unit vectors \(\mathbf{r}_{ic}=\mathbf{r}_{c}=\mathbf{r}_{i}=\mathbf{r}\). Inspecting the exponential form of quaternions [33], we have the following:  
\[\left(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\right)^{(1-\mu_{\omega} )}=\exp\left((1-\mu_{\omega})\log\overline{\mathbf{q}}_{w}^{{}^{\prime}}\right)= \exp((1-\mu_{\omega})[0,\theta\mathbf{r}])\ \, \tag{10}\] \[=\left[\cos((1-\mu_{\omega})\theta),\mathbf{r}\sin((1-\mu_{\omega}) \theta)\right]\] \[\left(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\right)^{\mu_{\omega}}= \exp\left(\mu_{\omega}\log\overline{\mathbf{q}}_{w}^{{}^{\prime}}\right)=\exp(\mu _{\omega}[0,\theta\mathbf{r}])\ \.\] (11) Three conditions are discussed as follows: 1. When \(\mu_{\omega}=1\), \(\left(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\right)^{0}=[1,0]\), and \(\left(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\right)^{1}=\left[\cos\theta,\mathbf{r} \sin\theta\right]\), \(\left(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\right)^{0}\otimes\left(\overline{ \mathbf{q}}_{w}^{{}^{\prime}}\right)^{1}=\left[\cos\theta,\mathbf{r}\sin\theta\right]= \overline{\mathbf{q}}_{w}^{{}^{\prime}}\). 2. When \(\mu_{\omega}=0\), similarly, \(\left(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\right)^{1}\otimes\left(\overline{ \mathbf{q}}_{w}^{{}^{\prime}}\right)^{0}=\overline{\mathbf{q}}_{w}^{{}^{\prime}}\). 3. When \(\mu_{\omega}\in(0,1)\), there is the following: \[\left(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\right)^{\mu_{\omega}}\otimes \left(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\right)^{(1-\mu_{\omega})}=\left[ \cos((1-\mu_{\omega})\theta),\mathbf{r}\sin((1-\mu_{\omega})\theta)\right][\cos( \mu_{\omega}\theta),\mathbf{r}\sin(\mu_{\omega}\theta)]\] \[=\left[\cos((1-\mu_{\omega})\theta)\cos(\mu_{\omega}\theta)-\sin ((1-\mu_{\omega})\theta)\sin(\mu_{\omega}\theta)(\mathbf{r}\cdot\mathbf{r}),\right.\] \[\left.\mathbf{r}\cos((1-\mu_{\omega})\theta)\sin(\mu_{\omega}\theta)+ \mathbf{r}\cos(\mu_{\omega}\theta)\sin((1-\mu_{\omega})\theta)+\left(\mathbf{r}\times \mathbf{r}\right)\sin((1-\mu_{\omega})\theta)\sin(\mu_{\omega}\theta)\right]\ \.\] (12) \[=\left[\cos\theta,\ \pi\sin\theta\right]=\overline{\mathbf{q}}_{w}^{{}^{ \prime}}\]  
Therefore, for all three conditions, \(\overline{\mathbf{q}}_{w}^{{}^{\prime}}=\left(\overline{\mathbf{q}}_{w}^{{}^{\prime}} \right)^{\mu_{\omega}}\otimes\left(\overline{\mathbf{q}}_{w}^{{}^{\prime}}\right)^ {1-\mu_{\omega}}\) holds. \(\square\)  
**Lemma A2**. : _Letting \(\mu\in[0,1]\), for the derivative of the \(\mu\)-th power of a general unit quaternion \(\overline{\mathbf{q}}\), \(\frac{d}{dt}(\overline{\mathbf{q}}^{\mu})=\frac{1}{2}\overline{\mathbf{q}}^{\mu} \otimes(\mu\overline{\mathbf{\omega}})\)._  
Proof of Lemma A2. : Referring to Equation (314) in [32] (pp.