An improved robotic arm constant force control method for forward terrain sensing system on planetary rovers

Abstract

The traversability of the terrain ahead of planetary rovers significantly impacts the success of their extraterrestrial exploration missions. Accurate perception of the terrain force through a forward wheel-on-limb detection system can provide crucial data for assessing the traversability of the terrain ahead. Existing constant force control methods, largely based on static terra mechanics models, struggle to meet the operational needs of planetary rovers in unknown environments. To address this issue, this paper proposes an improved adaptive impedance control method for robotic arms in unknown soft terrain. First, a dynamic soft terrain model is established to simulate the unknown soft terrain environment. Based on this model, an improved adaptive impedance control method is designed, and its stability is rigorously proven. The proposed method is then comprehensively validated through simulation experiments, hardware-in-the-loop tests, and experiments using a planetary rover model in flat, inclined, and curved terrain scenarios. Experimental results demonstrate the effectiveness of the designed adaptive impedance control method.

Introduction

The planetary rover, as a crucial vehicle for deep space exploration missions, often encounters challenges such as wheel sinkage and slippage during task execution. ¹ These issues significantly reduce its traversability on extraterrestrial planetary surfaces, consequently affecting the completion of deep space exploration missions. The forward wheel-on-limb detection system, represented by the wheeled bevameter, ² addresses the problem of visual blind spots inherent in traditional visual detection methods, making it a key focus of current research.

The forward wheel-on-limb detection device mounted on the planetary rover consists of a leg mechanism and a forward sensing wheel. The leg mechanism, often represented by a robotic arm, allows the device to follow the terrain and maintain constant force tracking control, while the forward sensing wheel collects real-time data on wheel-terrain interaction forces and torques, as shown in Figure 1. These data are crucial for assessing the traversability of the terrain ahead, providing necessary support for the rover's navigation decisions. Due to the extremely slow movement of planetary rovers (e.g., the Zhurong rover's average speed is approximately 1.1 cm/s), this forward wheel-on-limb detection equipment can thoroughly sense the wheel-terrain interaction information of the surface ahead, thereby providing real-time data for determining the traversability of the upcoming terrain. Research on forward wheel-on-limb detection devices has been explored by several scholars. Richter et al. ² first proposed the concept of a forward wheel-on-limb detection device in the European Space Agency's FASTER project, although the technical specifics were not disclosed. Zhang et al. ³ later proposed a wheel-on-limb device comprising a force-controlled robotic arm and a wheeled bevameter. The device was established to simulate traditional ground mechanics tests by applying vertical loads to the wheeled bevameter through the force-controlled robotic arm, with experiments validating the effectiveness of the proposed device. Feng et al. ⁴ developed a forward wheel-on-limb detection system featuring a linkage mechanism and a wheel force sensor for real-time terrain traversability assessment. The linkage mechanism facilitated terrain following, while the wheel force detector gathered the data of the real-time wheel-terrain interaction forces, with experiments confirming its effectiveness.

Figure 1.

Forward wheel-on-limb detection device.

The aforementioned studies suggest that the forward wheel-on-limb detection device represents the most promising solution for evaluating terrain traversability in planetary rover missions. In addition to accurately sensing the wheel-terrain interaction force, a key design requirement is the ability of the support leg mechanism to follow the terrain contours accurately while maintaining constant force tracking control. This requires a flexible robot arm that can follow the terrain in real-time and maintain constant force tracking on the forward sensing wheel to ensure the accurate acquisition of terrain data. Note that the reason for maintaining constant force tracking is to apply a fixed load to the forward sensing wheel through the robot arm to simulate the rover's traversal over the terrain ahead.

The implementation of a constant force control method for the forward wheel-on-limb detection device primarily involves designing corresponding control methods for leg mechanisms, represented by robotic arms. Common force and position control methods for robotic arms mainly include force/position hybrid control and impedance control. Force/position hybrid control achieves precise control of the robotic arm through specific trajectories (position control) and specific force feedback information when in contact with the environment. ⁵ Impedance control, on the other hand, realizes force and position control of the robotic arm through its dynamic characteristics. Although slightly less precise than force/position hybrid control, impedance control is more robust to variations in environmental conditions, making it well-suited for applications in uncertain environments.^6,7 Since the working environment of the forward wheel-on-limb detection system is an unknown environment, impedance control is adopted to achieve force and position control of the robotic arm. To improve the accuracy of impedance control, two mainstream improvement approaches are employed: the environmental parameter correction method and the variable impedance method. The environmental parameter correction method estimates external environmental parameters using existing information through designed methods. Jing et al. ⁸ proposed an adaptive dual-loop impedance control method, using a combination of inner and outer loops to achieve correction of environmental information, proving the effectiveness of the proposed method through experiments. Tsuji et al. ⁹ developed an online environmental stiffness parameter estimation method based on environmental noise changes, also proving successful through experimentation. Additionally, Roveda et al. ¹⁰ designed two extended Kalman filters to estimate interaction forces and environmental stiffness, validating the proposed control method by deploying it on the ‘Franka EMIKA Panda’ robotic hand. The variable impedance method adapts the impedance controller to unknown environmental stiffness and positions by designing impedance parameter adjustment rules. Hamedani et al. ¹¹ proposed an intelligent impedance control-based wavelet neural network for tracking dynamic contact forces in robotic arm position environments. This method achieves dynamic adjustment of impedance parameters through the online adaptive mechanism of the wavelet neural network, proving its effectiveness through simulation and physical experiments. Zhai et al. ¹² embedded neural networks into classic adaptive impedance controllers to optimize impedance control parameters, validating the proposed method through experiments. Additionally, many scholars have used methods such as iterative learning, ¹³ reinforcement learning, ¹⁴ and inverse reinforcement learning ¹⁵ to achieve adaptive adjustment of impedance parameters in unknown and variable environments.

Despite these studies on robotic arm impedance control methods, directly applying existing methods to unknown extraterrestrial soil conditions still presents the following challenges:

Most existing research is based on terrain mechanics models established for rigid environments, which do not apply to soft terrain scenarios.

Environmental parameters and terrain mechanics model parameters for soft terrains like soft terrains are largely unknown and non-static, making it difficult to establish accurate and reliable impedance models using methods that correct external environmental parameters.

Existing variable impedance control methods lack theoretical stability analysis for robotic arms under unknown soft terrain conditions, making them challenging to apply directly for constant force tracking control in such environments.

In this paper, we introduce an adaptive impedance control method specifically designed for robotic arm force control to achieve constant force tracking on soft terrain. Unlike existing control methods based on static terra mechanics models, our method first thoroughly considers the soft terrain environment and establishes a dynamic soft terrain model. Based on this terrain model, we design an adaptive impedance control method for the robotic arm to maintain constant force tracking. The stability of the proposed method is rigorously proven, and its effectiveness is demonstrated through a series of experiments.

The main contribution in this paper as follows:

Adaptive Impedance Control Method: We introduce an adaptive impedance control method specifically designed for robotic arms operating on unknown soft terrains, which improves the performance of traditional impedance methods.

Dynamic Soft Terrain Model: A dynamic soft terrain model has been developed to simulate the interaction between the rover's forward sensing wheel and soft terrain, enabling a more accurate prediction of wheel-terrain interaction forces.

Comprehensive Stability Analysis: The proposed method includes a rigorous theoretical stability analysis, ensuring reliable constant force tracking in unpredictable soft terrain environments.

The remainder of this paper is organized as follows: Section 2 details the design of a dynamic mechanical model for soft terrain and outlines the framework for constant force tracking of the forward wheel-on-limb detection device. Section 3 introduces the proposed adaptive variable impedance control method and provides a comprehensive stability analysis. Section 4 presents the simulation and physical experiments conducted to verify the effectiveness of the proposed constant force control method. Finally, section 5 concludes the paper with a summary of the method. For the convenience of reference, we have organized the symbols used in the text, as shown in Table 1.

Table 1.

Symbol description.

Symbol	Description
$z(t)$	Viscoelastic deformation of the soil at time $t$
$z_p(t)$	Plastic deformation of the soil at time $t$
$c_f$	Stiffness coefficient
$\eta$	damping coefficient
${\mu }_p$	Yield constants
${\mu }_v$	Compression coefficient
$p(t)$	Normal force of wheel-terrain interaction at time t.
$f_e$ , $f_e(t)$	Actual wheel-terrain interaction force
$f_d$ , $f_d(t)$	Desired wheel-terrain interaction force
$k_e$	Equivalent stiffness coefficient
$b_e$	Equivalent damping coefficient
$E$	Error between actual and desired the end of robot arm's trajectory, e is its one-dimensional representation
$X$	Actual robot arm's trajectory, x and $x_m(t)$ is its one-dimensional representation
$X_r$	Desired robot arm's trajectory, $x_r$ is its one-dimensional representation
$F$	Contact force
$K(s)$	Transfer function, $k(s)$ is its one-dimensional representation, s represents complex variable
$M$	Inertia matrix, m is its one-dimensional representation
$B$	Damping matrix, b is its one-dimensional representation
$K$	Stiffness matrix, k is its one-dimensional representation
$\mathrm{\Delta }F$	Deviation between actual and target force, $\mathrm{\Delta }f$ is its one-dimensional representation
$\mathrm{\Delta }{f}_{ss}$	The difference between the actual force and the the desired force of the robot arm in steady state
$x_e$	The position of the extraterrestrial soil surface
$\mathrm{\Delta }{x}_e$	The difference function between the actual planetary surface position and the expected planetary surface position
$\mathrm{\Delta }b(t)$	Equivalent damping function of the force deviation adjusted online
$\lambda$	Adjustment period of the controller
$\sigma$	The update rate of the parameters in the controller

Robot arm constant force tracking control framework

In this section, we first introduce the proposed constant force tracking control framework for the robotic arm in the forward wheel-on-limb detection device. Next, we present the proposed planetary terrain mechanics model. Finally, we discuss the common impedance control method and its associated challenges when applied to unknown soft terrain environments.

Constant Force Tracking Control Framework for Forward Wheel-On-Limb Detection Device

To achieve constant force tracking control of the forward wheel-on-limb detection device, we designed a control framework that adjusts the robotic arm's end position of the robotic arm based on the wheel-terrain contact force tracking feedback, as shown in Figure 2.

Figure 2.

Constant force tracking control framework of forward wheel-on-limb detection device.

The specific process of constant force tracking control described in Figure 2 is as follows:

The difference between $f_e$ and the current wheel-terrain contact force $f_d$ is input into the proposed adaptive impedance controller, resulting in the output position deviation e.

Superimpose the expected position $x_r$ of the robotic arm end and e to obtain the trajectory position x, which is then sent to the robotic arm position controller.

The robotic arm position controller employs a PD closed-loop control method with x as the input to control the movement of the arm's end-effector. Due to inherent errors in the system, the actual position of the end-effector is denoted as $x_m$ .

Using $x_m$ as input, the wheel-terrain contact force $f_e$ is calculated through a simulated soft terrain dynamic model. This model accounts for the complex interactions between the wheel and the soil surface, providing a realistic estimation of the contact force.

The above steps are repeated until e reaches a specified threshold, thereby achieving constant force tracking of the robotic arm.

Soft Terrain Modeling

Unlike traditional static terrain mechanics models with known terrain parameters, such as those by Bekker and Wong-Reece,^16,17 the soil parameters of soft terrain are unknown, and the terrain sensing process in front of the rover is dynamic. Therefore, based on the research of Gibbesch et al., ¹⁸ this paper constructs a dynamic mechanical model of soft terrain, as shown in Figure 3. In this model, $z(t)$ represents the viscoelastic deformation of the soil at time t, $z_p(t)$ represents the plastic deformation of the soil at time t, $c_f$ denotes the stiffness coefficient, $\eta$ represents the damping coefficient, ${\mu }_p$ represents yield constants, ${\mu }_v$ represents compression coefficient, and $p(t)$ represents the normal force of wheel-terrain interaction at time t. From Figure 3, we can derive the corresponding ground mechanics equation (1). By differentiating both sides of the second equation in (1) and substituting it into the first equation of (1), we obtain equation (2). Further solving equation (2) yields equation (3).

$$\{\begin{array}{c}p(t)=c_f(z(t)-z_p(t))+\eta (\dot{z}(t)-{\dot{z}}_p(t))\\ p(t)={\mu }_v{z}_p(t)+{\mu }_p\end{array}$$

(1)

$$\dot{p}(t)+{\displaystyle \frac{c_f+{\mu }_v}{\eta }p(t)={\mu }_v\dot{z}(t)+{\displaystyle \frac{c_f+{\mu }_v}{\eta }z(t)+{\displaystyle \frac{c_f+{\mu }_p}{\eta }}}}$$

(2)

$$\{\begin{array}{c}p(t)=p_1(t)p_2(t)+{\displaystyle \frac{c_f{\mu }_v}{c_f+{\mu }_v}z(t)}\\ p_1(t)={\left({\displaystyle \frac{{\mu }_v}{c_f+{\mu }_v}}\right)}^2\eta \dot{z}(t)+{\displaystyle \frac{c_f{\mu }_p}{c_f+{\mu }_v}}\\ p_2(t)=1-e^{(c_f+{\mu }_v/\eta )z(t)}\end{array}$$

(3)

Figure 3.

Dynamic mechanical model of soft terrain.

Due to the presence of nonlinear components in equation (3), it is challenging to use this equation for proving the stability of the subsequent constant force tracking control method. Therefore, it needs to be linearized. Considering that in a stable working state, the wheel sinkage no longer changes, i.e., $\dot{z}(t_0)=0$ , and the end position of the robotic arm at this time is $z(t_0)$ . The actual wheel-terrain interaction force $f_e$ equals the desired contact force $f_d$ , that is, $f_e(t_0)=f_d(t_0)$ . Thus:

$$f_e(t_0)={\displaystyle \frac{c_f{\mu }_p}{c_f+{\mu }_v}(1-e^{(c_f+\frac{{\mu }_v}{\eta })z(t_0)})+{\displaystyle \frac{c_f{\mu }_v}{c_f+{\mu }_v}z(t_0)}}$$

(4)

Applying Taylor expansion to equation (3) at $\dot{z}(t_0)$ and $z(t_0)$ , we obtain the calculation formula for the normal force of wheel-terrain interaction:

$$f_e(t)=A+k_{e}z(t)+b_e\dot{z}(t)$$

(5)

Where $k_e$ is the equivalent stiffness coefficient of the soft terrain, and $b_e$ is the equivalent damping coefficient of the soft terrain.

$$\{\begin{array}{c}A=\left({\displaystyle \frac{c_f{\mu }_p}{c_f+{\mu }_v}}\right)(1-e^{(c_f+{\mu }_v/\eta )z(t_0)})+{\displaystyle \frac{c_f{\mu }_v}{c_f+{\mu }_v}z(t_0)}\\ k_e={\displaystyle \frac{c_f{\mu }_v}{c_f+{\mu }_v}-\left({\displaystyle \frac{c_f{\mu }_p}{\eta }}\right)e^{(c_f+{\mu }_v/\eta )z(t_0)}}\\ b_e=\left[{\left({\displaystyle \frac{{\mu }_v}{c_f+{\mu }_v}}\right)}^2\eta \right](1-e^{(c_f+{\mu }_v/\eta )z(t_0)})\end{array}$$

(6)

Impedance Controller

As mentioned in the introduction, the impedance control method is optimal for constant force control of the forward wheel-on-limb detection device equipped on planetary rovers in soft terrain environments. As shown in Figure 4, the core concept of impedance control is to equate the robotic arm system to a desired impedance model, specifically a ‘mass-spring-damper’ model. The input to this model is the difference E between the actual trajectory X and the desired trajectory $X_r$ of the robotic arm's end-effector at a given moment. The output of the model is the contact force F generated when the end-effector interacts with the environment. $K(s)$ represents the transfer function of this model, where s denotes the complex variable in the Laplace transform. M, B, and K represent the inertia matrix, damping matrix, and stiffness matrix of the model, respectively. We can describe this model using equation (7), where $\mathrm{\Delta }F$ represents the deviation between the force experienced by the robotic arm in three spatial dimensions and the target force, while $\dot{E}$ and $\ddot{E}$ represent the first and second-order derivatives of E, respectively.

$$\{\begin{array}{c}M\ddot{E}+B\dot{E}+KE=\mathrm{\Delta }F\\ K(s)={\displaystyle \frac{1}{M{s}^2+Bs+K}}\end{array}$$

(7)

Figure 4.

Impedance control model.

For ease of analysis, we typically simplify the three-dimensional impedance model to a one-dimensional impedance model. In this simplified model, M, B, K, E and $\mathrm{\Delta }F$ are replaced by m, b, k, e and $\mathrm{\Delta }f$ , respectively, where $\mathrm{\Delta }f=f_e-f_d$ , $m>0$ and $b>$ 0, as shown in equation (8).

$$\{\begin{array}{c}m\ddot{e}+b\dot{e}+ke=\mathrm{\Delta }f\\ k(s)={\displaystyle \frac{1}{m{s}^2+bs+k}}\end{array}$$

(8)

The above traditional impedance control method cannot guarantee the performance of constant force tracking control for robotic arms on unknown rugged terrain. Therefore, in the next section, we will present a detailed introduction to the proposed constant force tracking control method for robotic arms based on adaptive variable impedance, along with its stability proof.

Proposed method

Adaptive Impedance Control Method

When implementing constant force control of a robotic arm in unknown extraterrestrial soil environments, we need to consider two key factors: the surface mechanical parameters of the soft terrain and the flatness of the soil surface. In real-world scenarios, both the ground mechanical parameters and the surface flatness of soft terrain are often unknown. To address this problem, we first introduce the adaptive impedance control method for flat soft terrain conditions. Building upon this, we then present the adaptive impedance control method for uneven soft terrain conditions.

Constant force control of robotic arm on soft terrain

The challenge of achieving constant force control of a robotic arm on a flat extraterrestrial soil surface arises from the unknown mechanical parameters of the soil, namely the equivalent stiffness coefficient $k_e$ and the equivalent damping coefficient $b_e$ . To address this problem, we explore an adaptive impedance control method starting from the steady-state error in force tracking. Based on equation (5), the force tracking error can be expressed as follows:

$$\mathrm{\Delta }f=A+k_e(x_e-x_m(t))+b_e{\displaystyle \frac{d(x_e-x_m(t))}{dt}-f_d}$$

(9)

Where $x_e$ represents the position of the extraterrestrial soil surface, $k_e>0$ , $b_e>0$ , and $x_m(t)$ represents the position of the robotic arm's end effector at time t. When the motion of the robotic arm stabilizes, ${\dot{x}}_m(t)=0$ , $x_r+e=x_m(t)$ . Combining equation (9) and equation (8), we obtain:

$$\begin{array}{rl}& \mathrm{\Delta }f=f_e-f_d\\ & =A+k_e[x_e-x_m(t)]+b_e\cdot {\dot{x}}_m(t)-f_d\\ & =A+k_e[x_e-x_m(t)]-f_d\\ & =A+k_e[x_e-(x_r+\mathrm{\Delta }f\cdot k(s))]-f_d\end{array}$$

(10)

Furthermore, when the motion of the robotic arm stabilizes, $s=0$ . Substituting the value of s into equation (10) yields the steady-state error in constant force tracking $\mathrm{\Delta }{f}_{ss}$ :

$$\mathrm{\Delta }{f}_{ss}={\displaystyle \frac{k}{k+k_e}[A+k_e(x_e-x_r)-f_d]}$$

(11)

From equation (11), we find that when the steady-state error $\mathrm{\Delta }{f}_{ss}$ in constant force tracking is zero, one of the following conditions must be met:

$$\{\begin{array}{c}k=0\\ x_r={\displaystyle \frac{A}{k_e}+x_e-{\displaystyle \frac{f_d}{k_e}}}\end{array}$$

(12)

Due to the unknown equivalent stiffness coefficient $k_e$ and the equivalent damping coefficient $b_e$ of the extraterrestrial soil, the desired stiffness k in the impedance controller is set to zero. Substituting the value of k into equation (8), we get the corresponding adaptive impedance control method:

$$\{\begin{array}{c}m\ddot{e}+b\dot{e}=\mathrm{\Delta }f\\ k(s)={\displaystyle \frac{1}{m{s}^2+bs}}\end{array}$$

(13)

Constant force control of robotic arm on uneven soft terrain

The adaptive impedance control method proposed in the previous section is only effective when the soft terrain surface is flat, i.e., when $x_e$ is constant. However, in practical situations, the soft terrain surface may have undulations, which means not only $k_e$ , $b_e$ are unknown, but $x_e$ changes over time. To address this problem, we define the difference function between the actual planetary surface position and the expected planetary surface position as $\mathrm{\Delta }{x}_e$ , which is a function of t. Thus, we have:

$$\mathrm{\Delta }f=m\left(\ddot{e}+\mathrm{\Delta }{\ddot{x}}_e\right)+b(\dot{e}+\mathrm{\Delta }{\dot{x}}_e)=m\ddot{\hat{e}}+b\dot{\hat{e}}$$

(14)

To ensure that the steady-state error of the robotic arm's constant force tracking is zero, we introduce adaptive variable impedance control to compensate for the time-varying error, i.e.,

$$\{\begin{array}{rl}\mathrm{\Delta }f=& m\ddot{\hat{e}}+(b+\mathrm{\Delta }b(t))\dot{\hat{e}}\\ \mathrm{\Delta }b(t)=& {\displaystyle \frac{b}{\dot{\hat{e}}(t)}\rho (t)}\\ \rho (t)=& \rho (t-\lambda )+\sigma {\displaystyle \frac{f_e(t-\lambda )-f_d(t-\lambda )}{b}}\end{array}$$

(15)

Where $\ddot{\hat{e}}=\ddot{e}+\mathrm{\Delta }{\ddot{x}}_e$ , $\mathrm{\Delta }b(t)$ represents the equivalent damping function of the force deviation adjusted online, $\lambda$ represents the adjustment period of the controller, usually $\lambda >0$ , and $\sigma$ represents the update rate of the parameters in the controller. If $\sigma$ is too large, it will cause oscillation and instability of the robotic arm; if $\sigma$ is too small, it will cause a delay in the response of the robotic arm. Next, we will prove the stability of equation (15) and simultaneously determine the range of values for $\sigma$ .

Stability Analysis of Constant Force Control System

In this section, we first introduce the state equation of the constant force control system for the robot arm. Subsequently, based on this state equation, we prove the stability of the designed constant force control system and determine the range of values for $\sigma$ .

Constant force control system state equation

In the unknown soft terrain surface, besides the difficulty in obtaining its mechanical parameters, the desired trajectory $x_r$ of the robotic arm in the forward sensing equipment is also challenging to accurately obtain. Therefore, at the initial state, we can replace the initial position $x_e$ of the star soil surface with $x_r$ . Thus, we have:

$$\{\begin{array}{c}z=x_m(t)-x_e\\ e=x_e-x_m(t)\end{array}$$

(16)

From equations (5) and (16), we obtain:

$$f_e=A-k_{e}e-b_e\dot{e}$$

(17)

For convenience in subsequent calculations, we first derive equation (15) and then second-order differentiate it, replacing $\dot{e}$ in $m\ddot{e}+b\dot{e}$ with $m\stackrel{⃛}{e}+b\ddot{e}$ , resulting in:

$$\{\begin{array}{c}m\ddot{e}+b\dot{e}=\left(m-b{\displaystyle \frac{b_e}{k_e}}\right)\ddot{e}-{\displaystyle \frac{b}{k_e}{\dot{f}}_e}\\ m\stackrel{⃛}{e}+b\ddot{e}=\left(b-m{\displaystyle \frac{k_e}{b_e}}\right)\ddot{e}-m{\displaystyle \frac{{\ddot{f}}_e}{b_e}}\end{array}$$

(18)

We rearrange equation (15) to obtain:

$$\{\begin{array}{c}{f}_e(t)-f_d(t)=m(\ddot{e}+\mathrm{\Delta }{\ddot{x}}_e)+b(\dot{e}+\mathrm{\Delta }{\dot{x}}_e)+b\rho (t)\\ -m\mathrm{\Delta }{\ddot{x}}_e-b\mathrm{\Delta }{\dot{x}}_e=m\ddot{e}+b\dot{e}+b\rho (t)-[f_e(t)-f_d(t)]\\ -m\mathrm{\Delta }{\stackrel{⃛}{x}}_e-b\mathrm{\Delta }{\ddot{x}}_e=\stackrel{⃛}{m}e+b\ddot{e}+b\dot{\rho }(t)-[{\dot{f}}_e(t)-{\dot{f}}_d(t)]\end{array}$$

(19)

Let ${\hat{f}}_e(t)=A+k_e\mathrm{\Delta }{x}_e+b_e\mathrm{\Delta }{\dot{x}}_e$ . Substituting the first and second parts of equation (18) into the second and third parts of equation (19) respectively, and adding the resulting equations, we get:

$$\{\begin{array}{c}{F}_0(t)=F_1(t)+F_2(t)+F_3(t)\\ F_0(t)=-m{\ddot{\hat{f}}}_e(t)-b{\dot{\hat{f}}}_e(t)\\ \begin{array}{c}{F}_1(t)=-m{\ddot{f}}_e(t)-b{\dot{f}}_e(t)+A{k}_{e}b\rho (t)\\ F_2(t)=-A{k}_e[f_e(t)-f_d(t)]+A{b}_{e}b\dot{\rho }(t)\\ F_3(t)=-A{b}_e[{\dot{f}}_e(t)-{\dot{f}}_d(t)]\end{array}\end{array}$$

(20)

Let $c(t)=f_d(t)-f_e(t)$ , $r(t)=f_d(t)-{\hat{f}}_e(t)$ . Simplifying equation (20) results in:

$$\{\begin{array}{c}{R}_0(t)=R_1(t)+R_2(t)+R_3(t)\\ R_0(t)=m\ddot{r}(t)+b\dot{r}(t)\\ R_1(t)=m\ddot{c}(t)+b\dot{c}(t)\\ R_2(t)=k_{e}b\rho (t)+k_{e}c(t)\\ R_3(t)=b_{e}b\dot{\rho }(t)+b_e\dot{c}(t)\end{array}$$

(21)

Substituting the third part of equation (15) and the initial condition $\rho (0)=0$ into equation (21), we obtain:

$$\{\begin{array}{c}{R}_0(t)=R_1(t)+{R^{\mathrm{\prime }}}_2(t)+{R^{\mathrm{\prime }}}_3(t)\\ R_0(t)=m\ddot{r}(t)+b\dot{r}(t)\\ R_1(t)=m\ddot{c}(t)+b\dot{c}(t)\\ {R^{\mathrm{\prime }}}_2(t)=k_{e}c(t)+k_{e}c(t)+k_e[\sigma c(t-n\lambda )+\cdots +\sigma c(t-\lambda )]\\ {R^{\mathrm{\prime }}}_3(t)=b_e\dot{c}(t)+b_e[\sigma \dot{c}(t-n\lambda )+\cdots +\sigma \dot{c}(t-\lambda )]\end{array}$$

(22)

Taking the Laplace transform of equation (22) gives:

$${\displaystyle \frac{c(s)}{r(s)}={\displaystyle \frac{m{s}^2+bs}{m{s}^2+bs+(k_e+b_{e}s)[1+\sigma (e^{-n\lambda s}+\cdots +e^{-\lambda s})]}}}$$

(23)

Assuming n is a sufficiently large number, then:

$${\sum }_{n=1}^{\mathrm{\infty }}{e}^{-n\lambda s}={\displaystyle \frac{e^{-\lambda s}}{1-e^{-\lambda s}}}$$

(24)

When the adjustment period $\lambda$ of the controller is sufficiently small, equation (23) can be rewritten as:

$${\displaystyle \frac{c(s)}{r(s)}={\displaystyle \frac{m{s}^2+bs}{m{s}^2+bs+(k_e+b_{e}s)\left(1+\sigma {\displaystyle \frac{1-\lambda s}{\lambda s}}\right)}}}$$

(25)

From equation (25), the characteristic equation of the constant force control system is:

$$\begin{array}{rl}\lambda m{s}^3+& [\lambda b+b_e\lambda (1-\sigma )]s^2+[b_e\sigma +k_e\lambda (1-\sigma )]s\\ +& k_e\sigma =0\end{array}$$

(26)

Stability proof of the constant force control system

When the input is a step function, with $r(s)=1/s$ , according to the final value theorem, the final value theorem can be used to obtain the force steady-state error of the closed-loop control system as follows:

$$e_{ss}=\underset{s\to 0}{\lim }sE(s)=\underset{s\to 0}{\lim }s(c(s)-r(s))$$

(27)

Combining with equation (25), we get:

$$e_{ss}=\underset{s\to 0}{\lim }s\left[{\displaystyle \frac{ms+b}{m{s}^2+bs+(k_e+b_{e}s)\left(1+\sigma {\displaystyle \frac{1-\lambda s}{\lambda s}}\right)}-{\displaystyle \frac{1}{s}}}\right]=-1$$

(28)

Further simplifying, we obtain:

$$\{\begin{array}{c}\underset{s\to 0}{\lim }sc(s)=0\\ \underset{t\to \mathrm{\infty }}{\lim }c(t)=\underset{t\to \mathrm{\infty }}{\lim }(f_d(t)-f_e(t))=0\end{array}$$

(29)

That is, when $t\to \mathrm{\infty }$ , $f_e\to f_d$ , the actual contact force converges to the desired contact force, and the steady-state tracking error of the force is zero.

Solution for the range of $\sigma$

Under the condition of stability in the constant force control system, the range of σ is determined using the Routh-Hurwitz criterion. Firstly, the Routh table constructed from equation (26) is shown in Table 2.

Here,

$$b_1={\displaystyle \frac{a_{12}{a}_{21}-a_{11}{a}_{22}}{a_{21}}>0}$$

(30)

According to the Routh-Hurwitz criterion, for the constant force control system to be stable, the coefficients in the first column of Table 2 must be positive, thus:

$$\{\begin{array}{c}\lambda m>0\\ \lambda b+b_e\lambda (1-\sigma )>0\\ k_e\sigma >0\\ {\displaystyle \frac{a_{12}{a}_{21}-a_{11}{a}_{22}}{a_{21}}>0}\end{array}$$

(31)

Table 2.

Routh-Hurwitz criterion table.

Power
$s^3$	$\lambda m$	$b_e\sigma +k_e\lambda (1-\sigma )$
$s^2$	$\lambda b+b_e\lambda (1-\sigma )$	$k_e\sigma$
$s^1$	$b_1$	0
$s^0$	$k_e\sigma$	0

Typically, for $m>0$ , b > 0, $\lambda >0$ , it follows that:

$$\{\begin{array}{c}\sigma <{\displaystyle \frac{k_e\lambda }{k_e\lambda -b_e}}\\ \sigma >0\\ [\lambda b+b_e\lambda (1-\sigma )][b_e\sigma +k_e\lambda (1-\sigma )]>k_e\sigma \lambda m\end{array}$$

(32)

Since the designed control system is intended for unknown soft terrain environments, the range of $\sigma$ when the system is stable is independent of the mechanical parameters of the soil. Therefore, by scaling equation (32), we obtain:

$$\{\begin{array}{c}\sigma \le 1<{\displaystyle \frac{b_e}{k_e\lambda -b_e}+1}\\ [\lambda b+b_e\lambda (1-\sigma )][b_e\sigma +k_e\lambda (1-\sigma )]>\\ k_e\lambda (1-\sigma )\lambda b\ge k_e\sigma \lambda m\end{array}$$

(33)

From equations (33) and (34), the range of $\sigma$ when the system is stable is:

$$0<\sigma \le {\displaystyle \frac{\lambda b}{m+\lambda b}}$$

(34)

Experimental verification

To validate the performance of the proposed adaptive impedance control method, simulation experiments were designed based on three scenarios: soft terrain surface as a flat plane, soft terrain surface as a downward slope, and soft terrain surface as a curved surface. Concurrently, semi-physical and physical model experiments were designed to further verify the performance of the proposed control method.

Simulation Experiment Verification

To validate the constant force tracking control performance of the robotic arm in unknown soft terrain environments, a Simulink simulation platform for constant force control of the robotic arm was constructed, as shown in Figure 5. This platform mainly consists of an adaptive impedance controller, a robotic arm position control module, a soft terrain surface mechanical parameter setting module, and a wheel-terrain contact force calculation module. The adaptive impedance controller is constructed using equation (15). The robotic arm position control module is used to simulate the end-effector control of the robotic arm. The soft terrain surface mechanical parameter setting module simulates the soft terrain surface by adjusting mechanical parameters. The wheel-terrain contact force calculation module is designed based on equation (10).

Figure 5.

Simulink simulation block diagram.

Due to the $4ms$ communication cycle between the robotic arm and the upper computer in the forward wheel-on-limb detection device, to ensure consistency between the simulation control effect and the actual robotic arm control effect, the simulation running cycle is also set to $4ms$ . Similarly, the desired contact force $f_d$ is set to $30N$ . The initial position of the robotic arm is set to 0 cm, the inertia coefficient $m=2.361\mathrm{kg}$ , the initial damping coefficient $b=135\mathrm{N}\cdot \mathrm{s}/\mathrm{m}$ , the stiffness coefficient $k=0\mathrm{N}/\mathrm{m}$ , and the parameter update rate $\sigma =0.03s^{-1}$ . In the current simulation experiment, as well as in the subsequent experiments, we employ a PD closed-loop control method as the position controller for the robotic arm's end-effector. This ensures precision in constant force tracking and enables accurate task execution during both simulation, semi-physical and physical tests.

Next, we will simulate and test the performance of the designed constant force tracking control method through different scenarios:

Flat soft terrain environment

Set the soft terrain surface position to 0 cm, and the initial position of the robotic arm end-effector to $z=-1\mathrm{cm}$ . Additionally, set the soft terrain environment parameters as follows:

$$\{\begin{array}{c}{k}_e=3000\mathrm{N}/\mathrm{m},b_e=200\mathrm{N}\cdot \mathrm{s}/\mathrm{m},0s\le t<4.3s\\ k_e=5000\mathrm{N}/\mathrm{m},b_e=400\mathrm{N}\cdot \mathrm{s}/\mathrm{m},4.3s\le t<6s\end{array}$$

(35)

Adaptive impedance control and traditional impedance control methods were applied to simulate constant force control for the robotic arm. The results are shown in Figure 6. In the initial phase, after a brief adjustment period, the robotic arm stabilized within the desired contact force range. The proposed adaptive impedance control method reached stability faster than the traditional impedance method during this stage. At around $4.3s$ , a change in the stiffness and damping parameters of the soft terrain caused an instantaneous variation in the contact force. The robotic arm then quickly adjusted through the control algorithm and stabilized within the desired contact force range. During this phase, the maximum steady-state force tracking error using the adaptive impedance control method was $0.03N$ , while the maximum error using the traditional impedance control method was $1.25N$ . Although the traditional impedance method also achieved constant force control, its impedance adjustment time was approximately $20\text{\%}$ slower than that of the proposed adaptive impedance control method.

Figure 6.

Variation of wheel-terrain contact force in flat simulation experiment.

Based on the above analysis, although both the traditional impedance control method and the proposed adaptive impedance control algorithm achieve impedance adjustment within $1s$ in the flat simulation environment, meeting the practical demand mentioned in the introduction where the rover moves at a speed of approximately $1.1cm/s$ , the proposed adaptive impedance control algorithm demonstrates clear advantages in terms of faster adjustment time and reduced force control error. These improvements ensure superior performance in achieving constant force control for the robotic arm in unknown flat soft terrain, compared to traditional impedance control methods.

Downward sloping soft terrain environment

The soft terrain surface position is set to $0cm$ , and the initial position of the robotic arm end-effector is set to $z=10\mathrm{cm}$ . The environmental stiffness coefficient $k_e$ is set to $3000N/m$ , and the environmental damping coefficient $b_e$ is set to $200N\cdot s/m$ . Constant force simulation control of the robotic arm was performed using both traditional impedance control and the proposed impedance control methods. The results are shown in Figure 7 and Table 3. For the sloping soft terrain environment, both control methods can achieve stable force-tracking effects. However, traditional impedance control exhibits a stable force tracking error of $2.4N$ . Although the adaptive impedance control method also shows a steady-state force tracking error, it is approximately one order of magnitude smaller than that of the traditional impedance control method.

Figure 7.

Variation of wheel-terrain contact force and robotic arm position in inclined plane simulation experiment.

Table 3.

Comparison table of control effects of inclined plane simulation experiments.

Control method	Impedance adjustment time	Steady-state error
Impedance control	$0.7s$	$1.4N$
Adaptive impedance control	$0.5s$	$0.11N$

Based on the above analysis, the designed adaptive impedance method demonstrates superior constant force control performance for the robotic arm in downward-sloping soft terrain environments compared to the traditional impedance method.

Curved soft terrain environment

The soft terrain surface position is set to $0cm$ , and the initial position of the robotic arm end-effector is set to $z=10\mathrm{cm}$ . The environmental stiffness coefficient $k_e$ is set to $3000N/m$ , and the environmental damping coefficient $b_e$ is set to 200 $N\cdot s/m$ . The force-tracking effects and robotic arm displacements for both traditional impedance control and adaptive variable impedance control are shown in Figure 8. Combined with the data in Table 4, it can be observed that as the soft terrain surface becomes more complex, both control methods exhibit small deviations around the desired contact force. However, the deviations of the adaptive variable impedance control are smaller compared to those of the traditional impedance control, resulting in more ideal control effects for the robotic arm.

Figure 8.

Variation of wheel-terrain contact force and robotic arm position in sinusoidal surface simulation experiment.

Table 4.

Comparison table of control effects of sinusoidal surface simulation experiments.

Control method	Impedance adjustment time	Steady-state error
Impedance control	$0.9s$	$1.5N$
Adaptive impedance control	$0.7s$	$0.61N$

In summary, for soft terrain environments with unknown stiffness and damping coefficients, including flat surfaces and downward slopes, the adaptive variable impedance control method can achieve stable force tracking effects. As the soft terrain surface becomes more complex, the adaptive variable impedance control also shows some force tracking deviations. However, compared to the traditional impedance method, the steady-state force deviations are smaller, and the constant force control performance of the robotic arm is superior.

Semi-physical Verification

To further validate the feasibility of the designed constant force control method for the robotic arm, we designed a semi-physical experiment based on the forward wheel-on-limb exploration system shown in Figure 9. This system consists of a robotic arm, slip ring, encoder, and forward sensing wheel. The robotic arm's end-effector is connected to the forward sensing wheel, which contains a multi-dimensional force sensor to collect wheel-terrain interaction forces in front of the planetary rover. The collected data is transmitted to the upper computer through the slip ring using CAN communication protocol with a $4ms$ communication cycle. The encoder provides real-time rotation angles of the forward sensing wheel for multi-dimensional force decoupling. Since this paper focuses on constant force control of the robotic arm, we will not discuss the specific force decoupling method in detail; readers can refer to existing research for more information.^19–21 The real-time position of the robotic arm is collected through its end-effector controller and transmitted to the upper computer via Socket communication.

Figure 9.

Semi-physical experimental platform.

To better simulate the constant force control scenario of the robotic arm during planetary rover exploration, we use a uniform lateral movement of the robotic arm to simulate its motion following the rover. The desired contact force $f_d$ is set to $30N$ , the inertia coefficient $m=1kg$ , the initial damping coefficient b is $35N\cdot s/m$ , the stiffness coefficient k is $0N\cdot s/m$ , and the parameter update rate $\sigma$ is $0.02{\mathrm{s}}^{-1}$ . The soft terrain surface position is set to $0cm$ . A soil bin experimental platform is used to simulate the soft terrain surface. The soil bin is $2m$ long, $0.5m$ wide, and $60cm$ high, filled with sand of different particle sizes mixed. Due to the random proportions of different particle sizes, this setup can simulate an unknown soft terrain environment.

Flat soft terrain testing

The mixed sand in the soil bin test platform was spread out evenly with a thickness of $30cm$ . The initial position of the robotic arm end-effector is set to $z=-2\mathrm{cm}$ . Initially, the bottom of the forward sensing wheel was lowered by the robotic arm to a depth of $2cm$ in the sand. At this point, the position data collected by the robotic arm's end-effector was calibrated and output through the upper computer program. The robotic arm moved the forward sensing wheel laterally at a uniform speed of about $1cm/s$ . The collected wheel-terrain contact force and the position changes of the robotic arm throughout the process as shown in Figure 10. Compared to the simulation experiments, the actual impedance adjustment time was slightly longer. Furthermore, during the stable phase, the designed adaptive impedance method showed smaller steady-state errors in force tracking compared to the traditional impedance method. For practical planetary exploration, a steady-state force error of less than $10\text{\%}$ is acceptable. ²² The adaptive impedance control method designed in this paper achieved a maximum actual steady-state force error of about 3%, while the traditional impedance method had a maximum steady-state force error of $7\text{\%}$ .

Figure 10.

Variation of wheel-terrain contact force and robotic arm displacement in planar experiment.

In summary, the designed adaptive impedance control method meets the actual precision requirements for collecting wheel-terrain interaction forces on flat soft terrain and outperforms the traditional impedance method.

Downward slope soft terrain testing

Fine sand was spread in the bin to form a slope for conducting the constant force-tracking experiment with the robotic arm. Figure 11 shows the wheel-terrain contact force and the position changes of the robotic arm. The initial position of the robotic arm end-effector is set to $z=10\mathrm{cm}$ . At the moment of contact, there was some overshoot. As the longitudinal displacement continued to adjust, the designed adaptive impedance control method reached a stable state in about $1.5s$ , while the traditional impedance method took about $2s$ to stabilize. Meanwhile, during the stable phase, the designed adaptive impedance control method had a maximum steady-state error in force tracking of about $5\text{\%}$ , while the traditional impedance method had a maximum steady-state error of about $8\text{\%}$ .

Figure 11.

Variation of wheel-terrain contact force and robotic arm displacement in inclined plane experiment.

In summary, compared to the flat environment, although the stability of constant force tracking by the designed adaptive impedance method for the robotic arm slightly decreased in the sloped environment, it still showed significant performance improvements compared to the traditional impedance control method.

Physical Verification

We designed a planetary rover model based on the original experimental platform to conduct on-site constant force tracking tests as shown in Figure 12. This device is powered by a $24V$ battery, has a maximum speed of $0.3m/s$ , and is controlled by a joystick.

Figure 12.

Forward terrain sensing system assembly in rover's model diagram.

The unknown soft terrain environment was simulated using random sandy terrain. The rover's constant speed was set to $2cm/s$ , with a desired contact force of $30N$ . The inertia coefficient m was set to $1kg$ , the initial damping coefficient b to $30N\cdot s/m$ , the stiffness coefficient k to $0N/m$ , and the parameter update rate $\sigma$ to $0.02{\mathrm{s}}^{-1}$ . The soft terrain surface position is set to $0cm$ , and the initial position of the robotic arm end-effector is set to $z=0\mathrm{cm}$ . The experimental results are shown in Figure 13. In the initial state, the damping adjustment time of the designed adaptive impedance control method was about $1.5s$ , while that of the traditional impedance control method was about $3s$ . Furthermore, during the stable phase, the maximum steady-state error in force tracking for the designed adaptive impedance method was $6\text{\%}$ , while for the traditional impedance method, it was $15\text{\%}$ . This indicates that the constant force control performance of the robotic arm using the designed adaptive impedance method is significantly superior to that of the traditional impedance method.

Figure 13.

The result of field tests of the designed control method and the traditional impedance control method.

Conclusion

The traversability of the terrain ahead of planetary rovers relies on an accurate perception of the forward terrain. The forward wheel-on-limb detection system, composed of a robotic arm and a forward sensing wheel, serves as crucial equipment for detecting terrain environment ahead of the rover. The stability of this system directly affects the prediction of terrain traversability. To address this problem, we have designed an improved adaptive constant force tracking control method for robotic arms in unknown soft terrain. First, considering the unknown characteristics of soft terrain mechanics parameters, we developed a dynamic terrain mechanical model. Based on this model, we designed an improved adaptive impedance control method for the robotic arm tailored to soft terrain and provided rigorous proof of its stability. Finally, we validated the proposed constant force control method through a comprehensive set of experiments, including simulation experiments, hardware-in-the-loop tests, and experiments using a planetary rover model in flat, inclined, and curved terrain scenarios. The experimental results demonstrate the effectiveness of the designed constant force control method. This paper contributes to enhancing the reliability and accuracy of terrain perception for planetary rovers, potentially improving the success rate of extraterrestrial exploration missions.