The kinematics defined in Eqs. 5–7 exploit the slip ratio and slip angle to calculate the longitudinal and lateral translational velocities. In this paper, we introduce four slip model types for estimating the slip ratio and slip angle, as mentioned in “Introduction” section.

For translational maneuvers, an approximated force model that considers resistance from terrain deformation has been introduced. Based on this force model, it is assumed that the resistances acting on each track are equivalent. Therefore, only the slip ratio and slip angle at the robot’s center of gravity need to be considered.

For turning maneuvers, the slip ratio is estimated by a previous method [7] that has been verified on a rigid plane; therefore, we confirmed its applicability on loose and weak slopes. On the other hand, a regression function is employed to estimate the slip angle. The regression function is trained by running data offline to identify the partial coefficients.

### Slippage during straight maneuvering

#### Force model of the robot

In order to estimate slippage during straight maneuvers, the interaction between the tracks and the terrain is accounted for. When a tracked vehicle travels along a straight line on weak ground, the vehicle tends to slip owing to motion resistance, as shown in Fig. 2. Resistance from the ground is mainly comprised of bulldozing and compaction resistances [10]. Bulldozing resistance is the force of pushing over or through the soil in front of the track. Compaction resistance, on the other hand, is the force of compacting the soil, which creates ruts in the terrain after the track has passed. In this research, it is assumed that majority of the soil ahead of the track is compacted, and thus, the bulldozing resistance has not been considered.

Compaction resistance is determined by calculating the work done in making a rut. The depth of the rut is called the sinkage, which is caused by the static weight of the robot and dynamic rotational motion of the trackbelt. Here, it is assumed that the robot moves with a low velocity, and the influence of dynamic sinkage on account of trackbelt rotation is small. Therefore, no dynamic sinkage has been considered in this study. The sinkage \(z_0\) and compaction resistance \(R_c\) are expressed as follows [10]:

$$\begin{aligned} z_0&= \root n' \of {\left( \frac{p}{k_c/b + k_\phi } \right) }, \end{aligned}$$

(8)

$$\begin{aligned} R_c&= b \int _0^{z_0} p dz, \end{aligned}$$

(9)

where \(k_c\), \(k_\phi\), and \(n'\) represent pressure-sinkage moduli of the soil based on the static sinkage model; *p* is the pressure on the bottom of the track; and *b* is the track width.

Gravitational force on the robot acts as tractive resistance, as shown in Fig. 2b and c. The longitudinal tractive resistance is parallel to the compaction resistance, whereas the lateral tractive resistance acts normal to the compaction resistance. These resistances are governed by the posture and weight of the robot as the following equation describe.

$$\begin{aligned} R_{t,long}&= (W\cos \theta _{roll}) \cdot \sin \theta _{pitch}, \end{aligned}$$

(10)

$$\begin{aligned} R_{t,lat}&= (W\cos \theta _{pitch}) \cdot \sin \theta _{roll}, \end{aligned}$$

(11)

Here, \(R_{t,long}\) represents longitudinal tractive resistance, \(R_{t,lat}\) is the lateral tractive resistance, *W* is weight of the track, and \(\theta _{roll}\) and \(\theta _{pitch}\) are roll and pitch angles of the track, respectively.

Therefore, the total resistance acting on the robot comprises the compaction and tractive resistances as follows:

$$\begin{aligned} R_{long}&= R_c + R_{t,long}, \end{aligned}$$

(12)

$$\begin{aligned} R_{lat}&= R_{t,lat}. \end{aligned}$$

(13)

The compaction and maximum tractive resistances acting on a climbing robot could be calculated by Eqs. 9 and 10, as shown in Fig. 3. The pressure-sinkage moduli used in this analysis correspond to those of sandy soil, snow, load [10], and lunar simulants with properties similar to volcanic ash [15]. Although tractive resistance increases with the slope angle, the compaction resistance remains largely independent with changes in the slope angle. It is, therefore reasonable to assume that compaction resistance \(R_c\) remains constant on homogeneous terrains.

#### Slip model for straight motion

Longitudinal slippage is expressed as a slip ratio, and lateral slippage is expressed as a slip angle. These slips can be derived using the force model and soil mechanics. To generate driving force, the track receives shear stress from the terrain equivalent to the corresponding driving force, as shown in Fig. 4. The corresponding shear stress \(\tau\) can be expressed with the following equation:

$$\begin{aligned} \tau (x) = \left( c + \frac{W}{bl} \tan \phi \right) \left( 1-e^{\frac{-j(x)}{K}} \right) , \end{aligned}$$

(14)

where *c* is the cohesion stress of the soil; *b* and *l* are the width and length of the track, respectively; \(\phi\) is the internal friction angle of the soil; *j*(*x*) is the shear displacement; and *K* is the shear deformation modulus. The adhesive force *c* represents adhesion between the soil and the track. Shear displacement *j*(*x*) is slippage at point *x* of the track, and can be described using slip ratio \(\alpha\) as follows:

$$\begin{aligned} j(x) = \alpha \cdot x. \end{aligned}$$

(15)

The shear deformation modulus *K* is determined by the slope of the shear stress curve at the origin. The value of *K* is defined by the interaction mechanics between the track and soil.

#### Slip ratio for straight motion

The driving force equals the longitudinal resistance for uniform linear motion; therefore, the driving force is derived to calculate the total shear stress beneath the track area. The equation for the driving force \(F_{drive}\) can be expressed as follows:

$$\begin{aligned} F_{drive}&= b \int ^{l}_{0} \tau (x) dx&\end{aligned}$$

(16)

$$\begin{aligned}&= \left( Ac + W\tan \phi \right) \left[ 1- \frac{K}{\alpha _{long} \cdot l} e^{\frac{-\alpha _{long} \cdot l}{K}} \right] ,\end{aligned}$$

(17)

where *A* is the contact area of the track. In a homogenous field, the cohesive stress and internal friction angle do not change; thus, the term on the left in Eq. 17 is constant. The term on the right in Eq. 17 can be expressed simply as a second-order approximation [16]. Under steady driving motion, the driving force and resistance are equivalent, and the slip ratio can be derived as follows:

$$\begin{aligned} F_{drive}&= R_{long}\end{aligned}$$

(18)

$$\begin{aligned}&= \left( Ac + W\tan \phi \right) \left[ 1- \frac{K}{\alpha _{long}l} e^{\frac{-\alpha _{long}l}{K}} \right]&\end{aligned}$$

(19)

$$\begin{aligned}&= C \cdot \alpha _{long},&\end{aligned}$$

(20)

$$\begin{aligned} \therefore \alpha _{long}&= \frac{R_{long}}{C} = \frac{R_c + R_{t,long}}{C}\end{aligned}$$

(21)

$$\begin{aligned}& \approx \frac{R_c}{C} + \frac{W}{C} \theta _{pitch}, \end{aligned}$$

(22)

where *C*, which replaces \(\left( \frac{Ac + W\tan \phi }{2K}l \right)\), has a constant value, and \(\alpha _{long}\) represents the slip ratio. If each track is subjected to the same resistance, the equivalent slip ratio \(\alpha _{long} = \alpha _r = \alpha _l\). By the small-angle approximation, the slip ratio is linear to the pitch angle of the robot and it has an intercept by the compaction resistance.

#### Slip angle during straight motion

The slip angle can be derived using a technique similar to that employed to obtain the slip ratio. The lateral resistance acting on the track is caused only by the weight of the robot (Eq. 13). The lateral slip ratio \(\alpha _y\) can then be written as follows:

$$\begin{aligned} \alpha _y = \frac{R_{lat}}{C'}, \end{aligned}$$

(23)

where \(C'\), which replaces \(\left( \frac{Ac + W\tan \phi }{2K'}l \right)\), has a constant value. The lateral transition velocity \(V_y\) in Eq. 2 can also be expressed in terms of the total lateral shear displacement \(j_y\) divided by contact time \(t_y\) as follows:

$$\begin{aligned} V_y = \frac{j_y}{t_y} = \frac{\alpha _y \cdot l}{\frac{l}{V_x}}=\alpha _y V_x, \end{aligned}$$

(24)

where \(t_y\) is the time at which a point on the track comes in contact with the terrain.

Finally, the following slip angle is derived from Eqs. 2, 23, and 24 using the small-angle approximation:

$$\begin{aligned} \beta&= \arctan { \left( \frac{V_y}{V_x} \right) } = \arctan { \left( \frac{R_{lat}}{C'} \right) } \end{aligned}$$

(25)

$$\begin{aligned} & \approx \frac{W}{C'} \theta _{roll}, \end{aligned}$$

(26)

### Slippage during turning maneuvers

The previous subsections describe the simplified slip ratio and slip angle models for steady straight motion based on force interactions on a weak terrain. In this section, we introduce a slippage estimation method for a robot rotating on a slope. Turning motions are complex phenomena that can cause longitudinal and lateral slippage. Furthermore, the slippage may differ at various points on the track. For instance, the lateral slippage at the front edge of the track is larger than that at its middle. The force model of turning motions is complex and the accurate prediction of its behavior requires several parameters. Therefore, it is difficult to use the force model for a real unknown field. In this research, we introduce a slippage estimation method based on a non-force model during turning maneuvers.

#### Slip ratio during turning maneuvers

To estimate the slip ratio during rotation, Endo et al. focused on the relationship between the slip ratio and input velocity of the track, and they proposed the following Eq. [7]:

$$\begin{aligned} \frac{\alpha _l}{\alpha _r} = - sgn \left( v_r \cdot v_l \right) \left( \left| \frac{v_r}{v_l} \right| \right) ^n, \end{aligned}$$

(27)

where *n* is a robot-dependent empirical slip parameter. If the angular velocity can be measured using sensors (i.e., gyroscopes), the angular velocity of the robot is equal to Eq. 7. Therefore, the slip ratio can be obtained from Eqs. 7 and 27. Parameter *n* is known to be constant on a horizontal solid plane. The applicability of this equation is confirmed in next section.

#### Slip angle during turning maneuvers

The lateral forces acting on the track during turning motion include not only the weight, as described for straight maneuvers, but also rotation resistance owing to the turning force, because a skid-steering track vehicle should slip to turn. The lateral forces consist of the bulldozing resistance acting on the sides of the tracks, shear resistance that rotates the robot and acts on the bottom of the tracks, and the weight of the robot acting parallel to the slope.

The bulldozing resistance is derived from soil parameters, rotation resistance, and angular displacement and posture of the robot. The shearing resistance is obtained from the rotation radius, angular displacement, and configuration of the robot. If the terrain is homogeneous, the values of variables required to derive the lateral force are obtained as internal information from the robot that can be directly measured by sensors, such as the IMU, and encoder. In this study, we introduce multiple linear regression function to estimate the slip angle, using the robot’s posture, yaw angular velocity, yaw angular displacement, slope angle, input velocity, and angular velocity as explanatory variables. Note that the angular displacement is the angle from the initial orientation of the turning movement. The regression function is defined as follows:

$$\begin{aligned} \beta = \sum \limits _{i=0}^{7} a_i X_i, \end{aligned}$$

(28)

where \(\beta\) is the predicted slip angle, \(X_i\) is *i*th explanatory variable, and \(a_i\) is the *i*th coefficient of the variable. \(X_0\) takes the value 1 and \(a_0\) is an intercept of the linear regression function. In addition, the slope angle \(\phi _{slope}\) can be derived from the posture of the robot as follows:

$$\begin{aligned} \phi _{slope} = \arccos \left( \cos \theta _{roll} \cdot \cos \theta _{pitch} \right) . \end{aligned}$$

(29)

The coefficients are identified using a training dataset to minimize the following squared error:

$$\begin{aligned} R = \sum \limits _{i=0}^{k} \left( B_i - \beta _i \right) ^2, \end{aligned}$$

(30)

where *k* is the number of training data points, \(B_i\) is the *i*th slip angle ground truth measured by an observation instrument, such as a motion capture camera, and \(\beta _i\) is the *i*th predicted slip angle using Eq. 28.