# Modular Motion-Structure Design Model for Planetary Surface Sampling

Authors: Qian Li; Lanlan Xie; Junping Li
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

Because there are less restrictions in space, a variety of different movement patterns and equipment structures may be used during the process of planetary surface sampling. Traditionally, the optimal analysis for surface sampling is focused on specific equipment structures and movements; in contrast, a new modular motion-structure design model for surface sampling, which is a more flexible model, is discussed in this paper. By establishing and combining two basic module groups, namely, the motion group and the structure group, this new design model can define and analyse multiple movement patterns and structures. For the motion group, calculating the sampling trajectory is the main purpose, in which there are two basic modules: tridimensional uniform rectilinear movements and tridimensional uniform circular movements. The two basic motion modules can be freely combined in a given coordinate system to simulate a random sampling trajectory. The structure group contains a series of curved and flat plates, which can be defined by a set of unified parameters (including section, extension, and cutting parameters). By assigning different values to these parameters, the curved or flat plates can represent different external shapes. The different structures of the various pieces of surface sampling equipment can be simulated by combining these different plates. In addition to defining these basic modules, analysing the coupling among different modules, which can be simplified to the relationship between velocity and surface, plays an important role in establishing this design model. Based on the modular design theory, this new model will not only reduce the difficulty of analysis but also improve accuracy for planetary surface sampling.


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## Body

## 1. Introduction

Research on extraterrestrial planets will be ongoing because of the ability of human beings to explore the universe. In the scientific field, this type of research can enrich the theories of the planets, the environment, the origin of life, and other related areas. Research on extraterrestrial planets can also bring significant developments in the areas of science, economy, military, etc. Since the Chinese lunar exploration project started in 2004, named Chang’e, the target of exploring the Moon in the first stage has been completed, and the second stage is focused on a lunar soft landing; an automatic inspection survey is now in progress. According to the research plan, the third stage of landing on the Moon, the sampling of lunar soil and return to Earth, will be completed sometime between 2017 and 2020 [1]. Meanwhile, the Chinese exploration of Mars will begin approximately in 2020, and this mission is expected to launch an exploratory rover to complete some preliminary investigations.To study extraterrestrial planets, sampling is the most direct and effective method; however, it is also a difficult method to implement. Planetary sampling methods are widely studied by many countries around the world. According to the difference of sampling depth, the planetary sampling can be divided into surface sampling and deep sampling, and the two sampling modes also require different sampling movements and equipment structures. For deep sampling, drilling is nearly the only way to obtain samples because of the limited movement space. For example, drilling was used in the lunar projects of Luna (Russia) and Apollo (U.S.A.), the Venusian project of Venera (Russia), the Martian projects of Deep Space 2 (U.S.A.) and Beagle 2 (EU), the cometary project of Rosetta (EU), etc. [2]. And in recent researches, some of the researchers also start to develop relative new sampling mode similar to drilling in order to increase sampling efficiency; for instance, the dual-reciprocating drill system developed by the University of Surrey can control the lateral movements and increase drilling depth [3, 4]. Compared with deep sampling, because there is hardly any space limitations, the sampling movement and equipment structure of surface sampling are more complex. According to the actual surface sampling projects for Moon, Mars, and Venus, a variety of sampling devices have been used to perform many tasks, such as digging, shovelling, clamping, grinding (to obtain the rock powder through grinding), and adsoring (to obtain dust through electrostatic adsorption), as shown in Figure 1 [2, 5–8]. Even for the same sampling movement, different equipment structures may be used; e.g., NASA had developed three different types of digging structures [8].Figure 1
Various surface sampling equipment structures: (a) digging, (b) shovelling, and (c) clamping.
(a)
(b)
(c)In the area of sampling devices research, there are two typical directions, one of the directions is focused on the entire systems which are considered as a robot. In this direction, the robots are usually researched by using the explicit mention of Lagrangian formalisms adapted to robotics. For instance, on the basis of the Kane equation that developed to describe the dynamic behavior for constrained multi-degree-of-freedom discrete system, the dynamic modelling and control method for a bicycle robot and hydraulic excavator have been developed by Beijing University of Posts and Telecommunications and University of Kragujevac, respectively [9, 10]. And according to research, Texas A&M University proposed analytical dynamics of variable-mass systems through the theory similar to the Lagrange equation, Appell equations, and Kane equations [11]. This direction pays much more attention on the entire system and therefore can hardly figure out the reaction force acting on the sampling devices.On the other hand, when the researches are focused on the sampling devices, either abroad or domestic, they are typically focused on the areas of specific movement and equipment structure. For instance, Carnegie Mellon University developed continuous excavation device, Jinlin University measured the torque using a specific scoop during sampling, and the University of California at Berkeley carried out several experiments for percussive excavation at different atmospheric pressures with a specially designed scoop [12–14]. This research approach limits the analysis accuracy and application. The various movements and structures increase the difficulty of theoretical analysis and design optimization in this area.To avoid the mentioned problems, a more flexible modular motion-structure design model for planetary surface sampling is discussed in this paper. This design model consists of two module groups, the motion and structure groups, and each group is composed of several basic modules (Figure2(b)). The basic modules from the same group should have the same parameter structure (Figure 2(a)). By combining these different basic modules, the new design model can be applied to design or analyse different movements and equipment structures (Figure 2(c)); as a result, this model is appropriate for the characteristics needed for planetary surface sampling.Figure 2
Modular motion-structure design model.


## 2. Motion Module Group

### 2.1. Motion Module Definition

The main function of the motion module group is to define the different motion patterns and draw a sampling trajectory for the surface sampling process. Based on the modular design concept, each basic module should be a separate part that has standard input and output parameters and an independent function. Hence, the sampling motion should be independent from the equipment structure, and the sampling equipment will be simplified to a particleP in the motion module group definition. The first step is to establish a coordinate system: a tridimensional coordinate system XYZ is built on the planetary surface, and the planetary ground plane is assumed to be plane XOY. Next, the motion modules are defined as mathematical models used to calculate the current coordinates of particle P at any moment in a given coordinate system, and the known conditions are the initial coordinates and the initial movement trend.To define the mathematical model of the motion modules, as shown in Figure3, assume the particle P is on the initial position A, where the coordinates are xst,yst,zst, and in the first phase, the relationship between velocity and time is V1t. After a certain time t1 of the movement, when the first phase is completed, the particle P moves to the position Bxmid,ymid,zmid and starts the second phase. In the second phase, which will last for an extended time t2, the relationship between velocity and time is V2t, and the particle P will reach position Cxend,yend,zend. During the period of t1+t2, the trajectory of particle P, which is shown as the thick solid line in Figure 3, is described by the mathematical model defined as equation (1). In this formula, the coordinates x,y,z of particle P at any time are the dependent variables, and the independent variables are the initial coordinates xst,yst,zst, the movement trend V1t,V2t, and time t. Equation (1) is the basic frame of the motion modules:
(1)x,y,z=S1Pst,V1t,t,t∈0,t1,S2Pmid,V2t,t,t∈t1,t2,where Pst is the initial coordinate of particle P in the first phase that corresponds to the coordinates of position Axst,yst,zst and Pmid is the initial coordinate of particle P in the second phase that corresponds to the coordinates of position Bxmid,ymid,zmid, Pmid=xmid,ymid,zmid=S1Pst,V1t1,t1.Figure 3
Motion module coordinate system.


### 2.2. Basic Modules in the Motion Group

From the perspective of speed, any motion with different accelerations can be divided into multiple ephemeral uniform motions. Regarding the trajectory, any tridimensional trajectory may be considered as the superposition of several rectilinear and circular trajectories. Hence, all tridimensional motions can be regarded as the combination of several tridimensional uniform rectilinear motions and circular motions, which can be defined as the basic modules for the motion group.


#### 2.2.1. Auxiliary Coordinate System for Motion Modules

To establish the trajectory equation of tridimensional uniform rectilinear motion and circular motion, an auxiliary coordinate system is required, as shown in Figure4. The auxiliary coordinate system is based on the global coordinate system XYZ shown in Figure 3, where the plane XOY is defined as the planetary surface. Assume the initial instantaneous velocity of sampling equipment is Vst, which is in the plane Au, and the initial position is O′. The angle between plane Au and plane XOY is θθ∈0,90, and the intersecting line is MN (assuming the initial position O′ is located on the intersecting line MN; i.e., all the sampling movements should start from the planetary surface, and the sampling initial position is the intersection position of the sampling equipment and the planetary surface). The angle ββ∈0,360 is defined as the angle between the intersecting line MN and axis X, and its positive direction is counterclockwise from the positive axis X. Thus, an auxiliary coordinate system X′O′Y′ is established on the plane Au, where the axis X′ is the intersecting line MN and axis Y′ is perpendicular to X′. Finally, define the angle αα∈0,360, which is the angle between initial velocity Vst and axis X′; its positive direction is counterclockwise from the positive axis X′.Figure 4
Auxiliary coordinate system for the motion module.


#### 2.2.2. Basic Module of Tridimensional Uniform Rectilinear Motion

In a three-dimensional space, the decomposition of the initial velocity vector is the core of uniform rectilinear motion trajectory analysis. Assuming the initial position isPstxst,yst,zst, the velocity is Vst and this movement will cost time t. According to the basic equation of uniform rectilinear motion, the end position Pendxend,yend,zend can be calculated as follows:
(2)Pend=Pst+Vstt.Bringing the initial velocityVst into the auxiliary coordinate system X′O′Y′, as shown in Figure 5(a), the velocity vector Vst can first be decomposed into vector Vx′xy and vector Vy′, as in equation (3). The vector Vx′xy is parallel to axis X′ and in the plane XOY, and the vector Vy′ is parallel to axis Y′.
(3)Vst→=Vx′xy→+Vy′→,Vx′xy=Vstcosα,Vy′=Vstsinα.Figure 5
Decomposition of the three-dimensional velocity vector.
(a)
(b)Second, decompose the vectorVy′ into vector Vy′xy and vector Vz, where vector Vy′xy is perpendicular to axis X′ and in the plane XOY and vector Vz is straight up:
(4)Vy′→=Vy′xy→+Vz→,Vy′xy=Vy′cosθ,Vz=Vy′sinθ.The vectorsVx′xy and Vy′xy are all in the plane XOY, and after transformation, the two vectors become the vector Vx and Vy, as shown in Figure 5(b):
(5)Vx→+Vy→=Vx′xy→+Vy′xy→,Vx=Vx′xycosβ+Vy′xycosβ−90=Vx′xycosβ+Vy′xysinβ,Vy=Vx′xysinβ+Vy′xysinβ−90=Vx′xysinβ−Vy′xycosβ.From equation (3) to equation (5), in three-dimensional coordinate system XYZ, the initial velocity vector Vst can be decomposed as three basic vectors along the corresponding axis X/Y/Z as follows:
(6)Vst→=Vx→+Vy→+Vz→,Vx=Vstcosαcosβ+sinαsinβcosθ,Vy=Vstcosαsinβ−sinαcosβcosθ,Vz=Vstsinαsinθ.According to equation (6) and equation (2), the basic mathematical module Sline of tridimensional uniform rectilinear motion is given by
(7)xend,yend,zend=Slinexst,yst,zst,Vst,α,β,θ,t,xend=xst+Vsttcosαcosβ+sinαsinβcosθ,yend=yst+Vsttcosαsinβ−sinαcosβcosθ,zend=zst+Vsttsinαsinθ.


#### 2.2.3. Basic Module of Tridimensional Uniform Circular Motion

Assume that there is a random particleP performing a tridimensional uniform circular movement, for which the radius is R, the initial position is Pstxst,yst,zst, and the initial linear speed is Vst. After a period of time t, particle P reaches the end position Pendxend,yend,zend. The solid line of Figure 6 presents the trajectory for particle P, and the angular velocity ωst can be calculated as
(8)ωst=Vst/R.Figure 6
Analysis of uniform circular motion in three dimensions.According to equation (8), when particle P reaches its end position, the deflection angle γ of initial linear velocity Vst is given by equation (9), and the positive direction of γ is counterclockwise from the positive axis X′
(9)γ=ωstt.Over the entire movement period, the value of initial linear vectorVst remains the same, while the direction changes. At the end position, decomposing the velocity Vst into velocity vector Vx″ parallel to axis X′ and vector Vy″ parallel to axis Y′ in the auxiliary coordinate system X′O′Y′, we have
(10)Vst→=Vx″→+Vy″→,Vx″=Vstcosα−γ,Vy″=Vstsinα−γ.If the circumference can be seen as consisting of multiple very short segments, assuming at the moment of timet, particle P maintains the same velocity (both value and direction) and continues to move in a very short time dt. At the end of period of dt, the displacement of the particle P is given by
(11)X″=Vx″dt,Y″=Vy″dt.Combining equation (8) to equation (11) and evaluating the entire period of time t, the total displacement in both directions of X′ and Y′ can be calculated as equation (12), in which the symbol ± represents the change of the motion direction.
(12)X′=∫0tX″=∫0tVstcosα±Vstt/Rdt=R∓sinα±sinα±Vstt/R,Y′=∫0tY″=∫0tVstsinα±Vstt/Rdt=R±cosα∓cosα±Vstt/R.According to equation (12), the end position PendX′,Y′, which is in the auxiliary coordinate system X′O′Y′, can be calculated. Moreover, through the mapping relationship between auxiliary coordinate system X′O′Y′ and initial coordinate system XYZ, the coordinates xend,yend,zend of the same position Pend can also be calculated. From equation (3) to equation (5), this mapping relationship is given as follows:
(13)xt=xt′cosβ+yt′sinβcosθ,yt=xt′sinβ−yt′cosβcosθ,zt=yt′sinθ.Bringing the results from equation (12) into equation (13), the basic mathematical module Scircle for tridimensional uniform circular motion is given as
(14)xend,yend,zend=Scirclexst,yst,zst,Vst,α,β,θ,t,R,xend=xst+R∓sinαcosβ±sinα±Vstt/Rcosβ±cosαsinβcosθ∓cosα±Vstt/Rsinβcosθ,yend=yst+R∓sinαsinβ±sinα±Vstt/Rsinβ∓cosαcosβcosθ±cosα±Vstt/Rcosβcosθ,zend=zst+R±cosαsinθ∓cosα±Vstt/Rsinθ.


### 2.3. Application of the Motion Group

#### 2.3.1. Calculation Procedure of the Motion Group

The main function of the motion group is to calculate the trajectory of the sampling movement by freely combining the basic modules from two groups. The trajectory drawing process is as follows:(1)
Standardizing the parameters of the basic modulesTo achieve the objective of freely combining modules, the two types of basic modules in the motion group should have the same types of input and output parameters. From the derivation of Section2.2, the uniform rectilinear module Sline and the uniform circular module Scircle do have the same output parameters outputS=xend,yend,zend. In terms of the input parameters, the parameter of the radius R is only in module Scircle. Hence, the radius R will be added into module Sline to ensure the unity of the input parameters. Theoretically, uniform rectilinear motion can be seen as a uniform circular motion with an infinite radius R=∞. In the actual calculation, in module Sline, the radius can be set as R=0, which indicates that the radius will not be involved in the calculation. More importantly, the parameter of radius R can be used to distinguish the rectilinear motion Sline and the circular motion Scircle. Thus, the same input parameters for both basic modules in the motion group can be obtained as InputS=xst,yst,zst,Vst,α,β,θ,t,R.
(2)
Creating the matrix of input parametersTo create the matrix of input parameters, first, division of the entire movement trajectory is necessary. The entire movement trajectory consists of several uniform rectilinear motions and circular motions, each of which will last different times, denoted ast1,t2,t3,…,tn. Next, the input parameters are extracted from each motion; combined with the duration time, the entire matrix MS is formed as given by
(15)MS=t1R1α1β1θ1Vst1t2R2α2β2θ2Vst2t3R3α3β3θ3Vst3⋮⋮⋮⋮⋮⋮tnRnαnβnθnVstn.Remarkably, the initial position for each motion is not involved in this matrix because in the whole calculation process, only the origin initial positionxst,yst,zst is required for the first motion. In the subsequent calculations, the initial position is the end position of the previous calculations, as shown in the following:
(16)xst_n,yst_n,zst_n=xend_n−1,yend_n−1,zend_n−1,n≥1.
(3)
Writing the related calculation programTo calculate movement trajectory, the program based on the basic motion group is comprised of four modules: the LINE module for calculating uniform rectilinear motion, the CIRCLE module for calculating uniform circle motion, the DRAW module for drawing trajectory, and the TRAJECTORY module for logic and data controlling. Figure7 shows the internal logic relationships among the four modules. To control the whole process, the first step of the TRAJECTORY module is to receive the external parameters (including the original initial position Pst, the input parameter matrix MS, and the entire movement time T). Next, the LINE module and the CIRCLE module are called in different periods according to the time division of parameter matrix MS. Every calculation for both modules will calculate a temporary end position Pend1,Pend2,Pend3,…. After completing all the calculations, all the temporary end positions can be integrated as an end position array Pends. Finally, to complete the entire calculation, the DRAW module will be called to draw the entire trajectory by using the end position array Pends. Figure 8 shows the detailed calculation process and the data flow of the whole program.Figure 7
Program modules with internal logic.Figure 8
Calculation process and data flow in the program.


#### 2.3.2. Example for Motion Group Application

Here, consider the example of the sampling movement of Phoenix Mars Lander, which landed on Mars on May 25th, 2008. To complete the sampling mission, a 2.35 m long Robotic Arm (RA) was used on the Phoenix Mars Lander; the RA had a 4-degree-of-freedom manipulator with a scoop at the top (Figure 9(a)). The scoop contained a front chamber and a rasp bit. The front chamber was designed to collect loose soil from the surface, and the rasp bit was designed for acquiring compact icy soil [15, 16].Figure 9
Phoenix Mars Lander Robotic Arm and its sampling process.
(a)
(b)
(c)
(d)The sampling process of the RA is shown in Figures9(b)–9(d). Figure 9(b) shows that the direction of sampling movement moved toward the Phoenix Lander itself. According to Figure 9(c) (where the photo was taken on the Phoenix) and Figure 9(d) (which shows the sampling trace), the movement trajectory of the front chamber in the sampling process can be divided into five stages:
(1)
Stage one. To reduce the resistance force from the surface soil, the chamber digs into the Martian surface at a small tilting angle. At this stage, the chamber is in a linear motion, and the digging will continue until the design sampling depth is reached
(2)
Stage two. The chamber is in a circular motion, and its forward direction will change from a tilted to a horizontal position
(3)
Stage three. The chamber continues to move along the horizontal direction and continues to collect the sample
(4)
Stage four. Until enough soil is collected, the chamber begins to perform circular movement; i.e., the forward direction will be changed back to the tilted position. However, the tilting angle will be much larger than that in stage one to prevent loss of the sample
(5)
Stage five. At the final stage, the chamber continues performing linear movements until it leaves the surfaceIn summary, the sampling trajectory for the digging chamber of the Phoenix Mars Lander can be defined as(17)SP=S1line,S2circle,S3line,S4circle,S5line.Assume that particleA is on the blade edge of the front chamber. To establish an initial coordinate system XYZ, the Mars surface can be set to plane XOY, and the origin O is set to the initial position of particle A. Because the actual movement trajectory parameters are undocumented, the trajectory calculation here is based on the assuming parameters as follows. For the whole sampling process, assume that the linear speed of the chamber remains the same at 5 mm/s, and the distance between particle A and rotation axis is 100 mm. For stage one, the tilting angle, which is defined as the angle between the digging direction and the surface, is set to 30°, which is assumed to provide the maximum insertion force. For stage three, the sampling depth is set to 10 cm, and the maximum horizontal displacement of RA is set to 500 mm. For stage four, to prevent sampling loss during the lifting process, the minimum tilting angle for the front chamber is set to 75°.According to the auxiliary coordinate system shown in Figure4, to simplify the calculation, the plane Au can be set to a vertical plane parallel to plane XOZ. Hence, the positional parameters are β=0 and θ=π/2 (in the computer calculation, the angle values must be in radians); thus, the input parameter matrix MSP for this assuming movement trajectory can be obtained, as given by the following:
(18)MSP=34.640−π/60π/2510.47100−π/60π/2539.30000π/2526.1810000π/255.3605π/120π/25.Introducing the matrixMSP into the program described in Figures 7 and 8, the assumed movement trajectory of particle A on the front chamber can be calculated, as shown by the solid line in Figure 10. If the width of the front chamber is 15 mm, then the trajectory of particle A can be moved to both sides by 7.5 cm to obtain the digging surface, as shown by the dashed line in Figure 10.Figure 10
Movement trajectory calculation model.


## 2.1. Motion Module Definition

The main function of the motion module group is to define the different motion patterns and draw a sampling trajectory for the surface sampling process. Based on the modular design concept, each basic module should be a separate part that has standard input and output parameters and an independent function. Hence, the sampling motion should be independent from the equipment structure, and the sampling equipment will be simplified to a particleP in the motion module group definition. The first step is to establish a coordinate system: a tridimensional coordinate system XYZ is built on the planetary surface, and the planetary ground plane is assumed to be plane XOY. Next, the motion modules are defined as mathematical models used to calculate the current coordinates of particle P at any moment in a given coordinate system, and the known conditions are the initial coordinates and the initial movement trend.To define the mathematical model of the motion modules, as shown in Figure3, assume the particle P is on the initial position A, where the coordinates are xst,yst,zst, and in the first phase, the relationship between velocity and time is V1t. After a certain time t1 of the movement, when the first phase is completed, the particle P moves to the position Bxmid,ymid,zmid and starts the second phase. In the second phase, which will last for an extended time t2, the relationship between velocity and time is V2t, and the particle P will reach position Cxend,yend,zend. During the period of t1+t2, the trajectory of particle P, which is shown as the thick solid line in Figure 3, is described by the mathematical model defined as equation (1). In this formula, the coordinates x,y,z of particle P at any time are the dependent variables, and the independent variables are the initial coordinates xst,yst,zst, the movement trend V1t,V2t, and time t. Equation (1) is the basic frame of the motion modules:
(1)x,y,z=S1Pst,V1t,t,t∈0,t1,S2Pmid,V2t,t,t∈t1,t2,where Pst is the initial coordinate of particle P in the first phase that corresponds to the coordinates of position Axst,yst,zst and Pmid is the initial coordinate of particle P in the second phase that corresponds to the coordinates of position Bxmid,ymid,zmid, Pmid=xmid,ymid,zmid=S1Pst,V1t1,t1.Figure 3
Motion module coordinate system.


## 2.2. Basic Modules in the Motion Group

From the perspective of speed, any motion with different accelerations can be divided into multiple ephemeral uniform motions. Regarding the trajectory, any tridimensional trajectory may be considered as the superposition of several rectilinear and circular trajectories. Hence, all tridimensional motions can be regarded as the combination of several tridimensional uniform rectilinear motions and circular motions, which can be defined as the basic modules for the motion group.


### 2.2.1. Auxiliary Coordinate System for Motion Modules

To establish the trajectory equation of tridimensional uniform rectilinear motion and circular motion, an auxiliary coordinate system is required, as shown in Figure4. The auxiliary coordinate system is based on the global coordinate system XYZ shown in Figure 3, where the plane XOY is defined as the planetary surface. Assume the initial instantaneous velocity of sampling equipment is Vst, which is in the plane Au, and the initial position is O′. The angle between plane Au and plane XOY is θθ∈0,90, and the intersecting line is MN (assuming the initial position O′ is located on the intersecting line MN; i.e., all the sampling movements should start from the planetary surface, and the sampling initial position is the intersection position of the sampling equipment and the planetary surface). The angle ββ∈0,360 is defined as the angle between the intersecting line MN and axis X, and its positive direction is counterclockwise from the positive axis X. Thus, an auxiliary coordinate system X′O′Y′ is established on the plane Au, where the axis X′ is the intersecting line MN and axis Y′ is perpendicular to X′. Finally, define the angle αα∈0,360, which is the angle between initial velocity Vst and axis X′; its positive direction is counterclockwise from the positive axis X′.Figure 4
Auxiliary coordinate system for the motion module.


### 2.2.2. Basic Module of Tridimensional Uniform Rectilinear Motion

In a three-dimensional space, the decomposition of the initial velocity vector is the core of uniform rectilinear motion trajectory analysis. Assuming the initial position isPstxst,yst,zst, the velocity is Vst and this movement will cost time t. According to the basic equation of uniform rectilinear motion, the end position Pendxend,yend,zend can be calculated as follows:
(2)Pend=Pst+Vstt.Bringing the initial velocityVst into the auxiliary coordinate system X′O′Y′, as shown in Figure 5(a), the velocity vector Vst can first be decomposed into vector Vx′xy and vector Vy′, as in equation (3). The vector Vx′xy is parallel to axis X′ and in the plane XOY, and the vector Vy′ is parallel to axis Y′.
(3)Vst→=Vx′xy→+Vy′→,Vx′xy=Vstcosα,Vy′=Vstsinα.Figure 5
Decomposition of the three-dimensional velocity vector.
(a)
(b)Second, decompose the vectorVy′ into vector Vy′xy and vector Vz, where vector Vy′xy is perpendicular to axis X′ and in the plane XOY and vector Vz is straight up:
(4)Vy′→=Vy′xy→+Vz→,Vy′xy=Vy′cosθ,Vz=Vy′sinθ.The vectorsVx′xy and Vy′xy are all in the plane XOY, and after transformation, the two vectors become the vector Vx and Vy, as shown in Figure 5(b):
(5)Vx→+Vy→=Vx′xy→+Vy′xy→,Vx=Vx′xycosβ+Vy′xycosβ−90=Vx′xycosβ+Vy′xysinβ,Vy=Vx′xysinβ+Vy′xysinβ−90=Vx′xysinβ−Vy′xycosβ.From equation (3) to equation (5), in three-dimensional coordinate system XYZ, the initial velocity vector Vst can be decomposed as three basic vectors along the corresponding axis X/Y/Z as follows:
(6)Vst→=Vx→+Vy→+Vz→,Vx=Vstcosαcosβ+sinαsinβcosθ,Vy=Vstcosαsinβ−sinαcosβcosθ,Vz=Vstsinαsinθ.According to equation (6) and equation (2), the basic mathematical module Sline of tridimensional uniform rectilinear motion is given by
(7)xend,yend,zend=Slinexst,yst,zst,Vst,α,β,θ,t,xend=xst+Vsttcosαcosβ+sinαsinβcosθ,yend=yst+Vsttcosαsinβ−sinαcosβcosθ,zend=zst+Vsttsinαsinθ.


### 2.2.3. Basic Module of Tridimensional Uniform Circular Motion

Assume that there is a random particleP performing a tridimensional uniform circular movement, for which the radius is R, the initial position is Pstxst,yst,zst, and the initial linear speed is Vst. After a period of time t, particle P reaches the end position Pendxend,yend,zend. The solid line of Figure 6 presents the trajectory for particle P, and the angular velocity ωst can be calculated as
(8)ωst=Vst/R.Figure 6
Analysis of uniform circular motion in three dimensions.According to equation (8), when particle P reaches its end position, the deflection angle γ of initial linear velocity Vst is given by equation (9), and the positive direction of γ is counterclockwise from the positive axis X′
(9)γ=ωstt.Over the entire movement period, the value of initial linear vectorVst remains the same, while the direction changes. At the end position, decomposing the velocity Vst into velocity vector Vx″ parallel to axis X′ and vector Vy″ parallel to axis Y′ in the auxiliary coordinate system X′O′Y′, we have
(10)Vst→=Vx″→+Vy″→,Vx″=Vstcosα−γ,Vy″=Vstsinα−γ.If the circumference can be seen as consisting of multiple very short segments, assuming at the moment of timet, particle P maintains the same velocity (both value and direction) and continues to move in a very short time dt. At the end of period of dt, the displacement of the particle P is given by
(11)X″=Vx″dt,Y″=Vy″dt.Combining equation (8) to equation (11) and evaluating the entire period of time t, the total displacement in both directions of X′ and Y′ can be calculated as equation (12), in which the symbol ± represents the change of the motion direction.
(12)X′=∫0tX″=∫0tVstcosα±Vstt/Rdt=R∓sinα±sinα±Vstt/R,Y′=∫0tY″=∫0tVstsinα±Vstt/Rdt=R±cosα∓cosα±Vstt/R.According to equation (12), the end position PendX′,Y′, which is in the auxiliary coordinate system X′O′Y′, can be calculated. Moreover, through the mapping relationship between auxiliary coordinate system X′O′Y′ and initial coordinate system XYZ, the coordinates xend,yend,zend of the same position Pend can also be calculated. From equation (3) to equation (5), this mapping relationship is given as follows:
(13)xt=xt′cosβ+yt′sinβcosθ,yt=xt′sinβ−yt′cosβcosθ,zt=yt′sinθ.Bringing the results from equation (12) into equation (13), the basic mathematical module Scircle for tridimensional uniform circular motion is given as
(14)xend,yend,zend=Scirclexst,yst,zst,Vst,α,β,θ,t,R,xend=xst+R∓sinαcosβ±sinα±Vstt/Rcosβ±cosαsinβcosθ∓cosα±Vstt/Rsinβcosθ,yend=yst+R∓sinαsinβ±sinα±Vstt/Rsinβ∓cosαcosβcosθ±cosα±Vstt/Rcosβcosθ,zend=zst+R±cosαsinθ∓cosα±Vstt/Rsinθ.


## 2.2.1. Auxiliary Coordinate System for Motion Modules

To establish the trajectory equation of tridimensional uniform rectilinear motion and circular motion, an auxiliary coordinate system is required, as shown in Figure4. The auxiliary coordinate system is based on the global coordinate system XYZ shown in Figure 3, where the plane XOY is defined as the planetary surface. Assume the initial instantaneous velocity of sampling equipment is Vst, which is in the plane Au, and the initial position is O′. The angle between plane Au and plane XOY is θθ∈0,90, and the intersecting line is MN (assuming the initial position O′ is located on the intersecting line MN; i.e., all the sampling movements should start from the planetary surface, and the sampling initial position is the intersection position of the sampling equipment and the planetary surface). The angle ββ∈0,360 is defined as the angle between the intersecting line MN and axis X, and its positive direction is counterclockwise from the positive axis X. Thus, an auxiliary coordinate system X′O′Y′ is established on the plane Au, where the axis X′ is the intersecting line MN and axis Y′ is perpendicular to X′. Finally, define the angle αα∈0,360, which is the angle between initial velocity Vst and axis X′; its positive direction is counterclockwise from the positive axis X′.Figure 4
Auxiliary coordinate system for the motion module.


## 2.2.2. Basic Module of Tridimensional Uniform Rectilinear Motion

In a three-dimensional space, the decomposition of the initial velocity vector is the core of uniform rectilinear motion trajectory analysis. Assuming the initial position isPstxst,yst,zst, the velocity is Vst and this movement will cost time t. According to the basic equation of uniform rectilinear motion, the end position Pendxend,yend,zend can be calculated as follows:
(2)Pend=Pst+Vstt.Bringing the initial velocityVst into the auxiliary coordinate system X′O′Y′, as shown in Figure 5(a), the velocity vector Vst can first be decomposed into vector Vx′xy and vector Vy′, as in equation (3). The vector Vx′xy is parallel to axis X′ and in the plane XOY, and the vector Vy′ is parallel to axis Y′.
(3)Vst→=Vx′xy→+Vy′→,Vx′xy=Vstcosα,Vy′=Vstsinα.Figure 5
Decomposition of the three-dimensional velocity vector.
(a)
(b)Second, decompose the vectorVy′ into vector Vy′xy and vector Vz, where vector Vy′xy is perpendicular to axis X′ and in the plane XOY and vector Vz is straight up:
(4)Vy′→=Vy′xy→+Vz→,Vy′xy=Vy′cosθ,Vz=Vy′sinθ.The vectorsVx′xy and Vy′xy are all in the plane XOY, and after transformation, the two vectors become the vector Vx and Vy, as shown in Figure 5(b):
(5)Vx→+Vy→=Vx′xy→+Vy′xy→,Vx=Vx′xycosβ+Vy′xycosβ−90=Vx′xycosβ+Vy′xysinβ,Vy=Vx′xysinβ+Vy′xysinβ−90=Vx′xysinβ−Vy′xycosβ.From equation (3) to equation (5), in three-dimensional coordinate system XYZ, the initial velocity vector Vst can be decomposed as three basic vectors along the corresponding axis X/Y/Z as follows:
(6)Vst→=Vx→+Vy→+Vz→,Vx=Vstcosαcosβ+sinαsinβcosθ,Vy=Vstcosαsinβ−sinαcosβcosθ,Vz=Vstsinαsinθ.According to equation (6) and equation (2), the basic mathematical module Sline of tridimensional uniform rectilinear motion is given by
(7)xend,yend,zend=Slinexst,yst,zst,Vst,α,β,θ,t,xend=xst+Vsttcosαcosβ+sinαsinβcosθ,yend=yst+Vsttcosαsinβ−sinαcosβcosθ,zend=zst+Vsttsinαsinθ.


## 2.2.3. Basic Module of Tridimensional Uniform Circular Motion

Assume that there is a random particleP performing a tridimensional uniform circular movement, for which the radius is R, the initial position is Pstxst,yst,zst, and the initial linear speed is Vst. After a period of time t, particle P reaches the end position Pendxend,yend,zend. The solid line of Figure 6 presents the trajectory for particle P, and the angular velocity ωst can be calculated as
(8)ωst=Vst/R.Figure 6
Analysis of uniform circular motion in three dimensions.According to equation (8), when particle P reaches its end position, the deflection angle γ of initial linear velocity Vst is given by equation (9), and the positive direction of γ is counterclockwise from the positive axis X′
(9)γ=ωstt.Over the entire movement period, the value of initial linear vectorVst remains the same, while the direction changes. At the end position, decomposing the velocity Vst into velocity vector Vx″ parallel to axis X′ and vector Vy″ parallel to axis Y′ in the auxiliary coordinate system X′O′Y′, we have
(10)Vst→=Vx″→+Vy″→,Vx″=Vstcosα−γ,Vy″=Vstsinα−γ.If the circumference can be seen as consisting of multiple very short segments, assuming at the moment of timet, particle P maintains the same velocity (both value and direction) and continues to move in a very short time dt. At the end of period of dt, the displacement of the particle P is given by
(11)X″=Vx″dt,Y″=Vy″dt.Combining equation (8) to equation (11) and evaluating the entire period of time t, the total displacement in both directions of X′ and Y′ can be calculated as equation (12), in which the symbol ± represents the change of the motion direction.
(12)X′=∫0tX″=∫0tVstcosα±Vstt/Rdt=R∓sinα±sinα±Vstt/R,Y′=∫0tY″=∫0tVstsinα±Vstt/Rdt=R±cosα∓cosα±Vstt/R.According to equation (12), the end position PendX′,Y′, which is in the auxiliary coordinate system X′O′Y′, can be calculated. Moreover, through the mapping relationship between auxiliary coordinate system X′O′Y′ and initial coordinate system XYZ, the coordinates xend,yend,zend of the same position Pend can also be calculated. From equation (3) to equation (5), this mapping relationship is given as follows:
(13)xt=xt′cosβ+yt′sinβcosθ,yt=xt′sinβ−yt′cosβcosθ,zt=yt′sinθ.Bringing the results from equation (12) into equation (13), the basic mathematical module Scircle for tridimensional uniform circular motion is given as
(14)xend,yend,zend=Scirclexst,yst,zst,Vst,α,β,θ,t,R,xend=xst+R∓sinαcosβ±sinα±Vstt/Rcosβ±cosαsinβcosθ∓cosα±Vstt/Rsinβcosθ,yend=yst+R∓sinαsinβ±sinα±Vstt/Rsinβ∓cosαcosβcosθ±cosα±Vstt/Rcosβcosθ,zend=zst+R±cosαsinθ∓cosα±Vstt/Rsinθ.


## 2.3. Application of the Motion Group

### 2.3.1. Calculation Procedure of the Motion Group

The main function of the motion group is to calculate the trajectory of the sampling movement by freely combining the basic modules from two groups. The trajectory drawing process is as follows:(1)
Standardizing the parameters of the basic modulesTo achieve the objective of freely combining modules, the two types of basic modules in the motion group should have the same types of input and output parameters. From the derivation of Section2.2, the uniform rectilinear module Sline and the uniform circular module Scircle do have the same output parameters outputS=xend,yend,zend. In terms of the input parameters, the parameter of the radius R is only in module Scircle. Hence, the radius R will be added into module Sline to ensure the unity of the input parameters. Theoretically, uniform rectilinear motion can be seen as a uniform circular motion with an infinite radius R=∞. In the actual calculation, in module Sline, the radius can be set as R=0, which indicates that the radius will not be involved in the calculation. More importantly, the parameter of radius R can be used to distinguish the rectilinear motion Sline and the circular motion Scircle. Thus, the same input parameters for both basic modules in the motion group can be obtained as InputS=xst,yst,zst,Vst,α,β,θ,t,R.
(2)
Creating the matrix of input parametersTo create the matrix of input parameters, first, division of the entire movement trajectory is necessary. The entire movement trajectory consists of several uniform rectilinear motions and circular motions, each of which will last different times, denoted ast1,t2,t3,…,tn. Next, the input parameters are extracted from each motion; combined with the duration time, the entire matrix MS is formed as given by
(15)MS=t1R1α1β1θ1Vst1t2R2α2β2θ2Vst2t3R3α3β3θ3Vst3⋮⋮⋮⋮⋮⋮tnRnαnβnθnVstn.Remarkably, the initial position for each motion is not involved in this matrix because in the whole calculation process, only the origin initial positionxst,yst,zst is required for the first motion. In the subsequent calculations, the initial position is the end position of the previous calculations, as shown in the following:
(16)xst_n,yst_n,zst_n=xend_n−1,yend_n−1,zend_n−1,n≥1.
(3)
Writing the related calculation programTo calculate movement trajectory, the program based on the basic motion group is comprised of four modules: the LINE module for calculating uniform rectilinear motion, the CIRCLE module for calculating uniform circle motion, the DRAW module for drawing trajectory, and the TRAJECTORY module for logic and data controlling. Figure7 shows the internal logic relationships among the four modules. To control the whole process, the first step of the TRAJECTORY module is to receive the external parameters (including the original initial position Pst, the input parameter matrix MS, and the entire movement time T). Next, the LINE module and the CIRCLE module are called in different periods according to the time division of parameter matrix MS. Every calculation for both modules will calculate a temporary end position Pend1,Pend2,Pend3,…. After completing all the calculations, all the temporary end positions can be integrated as an end position array Pends. Finally, to complete the entire calculation, the DRAW module will be called to draw the entire trajectory by using the end position array Pends. Figure 8 shows the detailed calculation process and the data flow of the whole program.Figure 7
Program modules with internal logic.Figure 8
Calculation process and data flow in the program.


### 2.3.2. Example for Motion Group Application

Here, consider the example of the sampling movement of Phoenix Mars Lander, which landed on Mars on May 25th, 2008. To complete the sampling mission, a 2.35 m long Robotic Arm (RA) was used on the Phoenix Mars Lander; the RA had a 4-degree-of-freedom manipulator with a scoop at the top (Figure 9(a)). The scoop contained a front chamber and a rasp bit. The front chamber was designed to collect loose soil from the surface, and the rasp bit was designed for acquiring compact icy soil [15, 16].Figure 9
Phoenix Mars Lander Robotic Arm and its sampling process.
(a)
(b)
(c)
(d)The sampling process of the RA is shown in Figures9(b)–9(d). Figure 9(b) shows that the direction of sampling movement moved toward the Phoenix Lander itself. According to Figure 9(c) (where the photo was taken on the Phoenix) and Figure 9(d) (which shows the sampling trace), the movement trajectory of the front chamber in the sampling process can be divided into five stages:
(1)
Stage one. To reduce the resistance force from the surface soil, the chamber digs into the Martian surface at a small tilting angle. At this stage, the chamber is in a linear motion, and the digging will continue until the design sampling depth is reached
(2)
Stage two. The chamber is in a circular motion, and its forward direction will change from a tilted to a horizontal position
(3)
Stage three. The chamber continues to move along the horizontal direction and continues to collect the sample
(4)
Stage four. Until enough soil is collected, the chamber begins to perform circular movement; i.e., the forward direction will be changed back to the tilted position. However, the tilting angle will be much larger than that in stage one to prevent loss of the sample
(5)
Stage five. At the final stage, the chamber continues performing linear movements until it leaves the surfaceIn summary, the sampling trajectory for the digging chamber of the Phoenix Mars Lander can be defined as(17)SP=S1line,S2circle,S3line,S4circle,S5line.Assume that particleA is on the blade edge of the front chamber. To establish an initial coordinate system XYZ, the Mars surface can be set to plane XOY, and the origin O is set to the initial position of particle A. Because the actual movement trajectory parameters are undocumented, the trajectory calculation here is based on the assuming parameters as follows. For the whole sampling process, assume that the linear speed of the chamber remains the same at 5 mm/s, and the distance between particle A and rotation axis is 100 mm. For stage one, the tilting angle, which is defined as the angle between the digging direction and the surface, is set to 30°, which is assumed to provide the maximum insertion force. For stage three, the sampling depth is set to 10 cm, and the maximum horizontal displacement of RA is set to 500 mm. For stage four, to prevent sampling loss during the lifting process, the minimum tilting angle for the front chamber is set to 75°.According to the auxiliary coordinate system shown in Figure4, to simplify the calculation, the plane Au can be set to a vertical plane parallel to plane XOZ. Hence, the positional parameters are β=0 and θ=π/2 (in the computer calculation, the angle values must be in radians); thus, the input parameter matrix MSP for this assuming movement trajectory can be obtained, as given by the following:
(18)MSP=34.640−π/60π/2510.47100−π/60π/2539.30000π/2526.1810000π/255.3605π/120π/25.Introducing the matrixMSP into the program described in Figures 7 and 8, the assumed movement trajectory of particle A on the front chamber can be calculated, as shown by the solid line in Figure 10. If the width of the front chamber is 15 mm, then the trajectory of particle A can be moved to both sides by 7.5 cm to obtain the digging surface, as shown by the dashed line in Figure 10.Figure 10
Movement trajectory calculation model.


## 2.3.1. Calculation Procedure of the Motion Group

The main function of the motion group is to calculate the trajectory of the sampling movement by freely combining the basic modules from two groups. The trajectory drawing process is as follows:(1)
Standardizing the parameters of the basic modulesTo achieve the objective of freely combining modules, the two types of basic modules in the motion group should have the same types of input and output parameters. From the derivation of Section2.2, the uniform rectilinear module Sline and the uniform circular module Scircle do have the same output parameters outputS=xend,yend,zend. In terms of the input parameters, the parameter of the radius R is only in module Scircle. Hence, the radius R will be added into module Sline to ensure the unity of the input parameters. Theoretically, uniform rectilinear motion can be seen as a uniform circular motion with an infinite radius R=∞. In the actual calculation, in module Sline, the radius can be set as R=0, which indicates that the radius will not be involved in the calculation. More importantly, the parameter of radius R can be used to distinguish the rectilinear motion Sline and the circular motion Scircle. Thus, the same input parameters for both basic modules in the motion group can be obtained as InputS=xst,yst,zst,Vst,α,β,θ,t,R.
(2)
Creating the matrix of input parametersTo create the matrix of input parameters, first, division of the entire movement trajectory is necessary. The entire movement trajectory consists of several uniform rectilinear motions and circular motions, each of which will last different times, denoted ast1,t2,t3,…,tn. Next, the input parameters are extracted from each motion; combined with the duration time, the entire matrix MS is formed as given by
(15)MS=t1R1α1β1θ1Vst1t2R2α2β2θ2Vst2t3R3α3β3θ3Vst3⋮⋮⋮⋮⋮⋮tnRnαnβnθnVstn.Remarkably, the initial position for each motion is not involved in this matrix because in the whole calculation process, only the origin initial positionxst,yst,zst is required for the first motion. In the subsequent calculations, the initial position is the end position of the previous calculations, as shown in the following:
(16)xst_n,yst_n,zst_n=xend_n−1,yend_n−1,zend_n−1,n≥1.
(3)
Writing the related calculation programTo calculate movement trajectory, the program based on the basic motion group is comprised of four modules: the LINE module for calculating uniform rectilinear motion, the CIRCLE module for calculating uniform circle motion, the DRAW module for drawing trajectory, and the TRAJECTORY module for logic and data controlling. Figure7 shows the internal logic relationships among the four modules. To control the whole process, the first step of the TRAJECTORY module is to receive the external parameters (including the original initial position Pst, the input parameter matrix MS, and the entire movement time T). Next, the LINE module and the CIRCLE module are called in different periods according to the time division of parameter matrix MS. Every calculation for both modules will calculate a temporary end position Pend1,Pend2,Pend3,…. After completing all the calculations, all the temporary end positions can be integrated as an end position array Pends. Finally, to complete the entire calculation, the DRAW module will be called to draw the entire trajectory by using the end position array Pends. Figure 8 shows the detailed calculation process and the data flow of the whole program.Figure 7
Program modules with internal logic.Figure 8
Calculation process and data flow in the program.


## 2.3.2. Example for Motion Group Application

Here, consider the example of the sampling movement of Phoenix Mars Lander, which landed on Mars on May 25th, 2008. To complete the sampling mission, a 2.35 m long Robotic Arm (RA) was used on the Phoenix Mars Lander; the RA had a 4-degree-of-freedom manipulator with a scoop at the top (Figure 9(a)). The scoop contained a front chamber and a rasp bit. The front chamber was designed to collect loose soil from the surface, and the rasp bit was designed for acquiring compact icy soil [15, 16].Figure 9
Phoenix Mars Lander Robotic Arm and its sampling process.
(a)
(b)
(c)
(d)The sampling process of the RA is shown in Figures9(b)–9(d). Figure 9(b) shows that the direction of sampling movement moved toward the Phoenix Lander itself. According to Figure 9(c) (where the photo was taken on the Phoenix) and Figure 9(d) (which shows the sampling trace), the movement trajectory of the front chamber in the sampling process can be divided into five stages:
(1)
Stage one. To reduce the resistance force from the surface soil, the chamber digs into the Martian surface at a small tilting angle. At this stage, the chamber is in a linear motion, and the digging will continue until the design sampling depth is reached
(2)
Stage two. The chamber is in a circular motion, and its forward direction will change from a tilted to a horizontal position
(3)
Stage three. The chamber continues to move along the horizontal direction and continues to collect the sample
(4)
Stage four. Until enough soil is collected, the chamber begins to perform circular movement; i.e., the forward direction will be changed back to the tilted position. However, the tilting angle will be much larger than that in stage one to prevent loss of the sample
(5)
Stage five. At the final stage, the chamber continues performing linear movements until it leaves the surfaceIn summary, the sampling trajectory for the digging chamber of the Phoenix Mars Lander can be defined as(17)SP=S1line,S2circle,S3line,S4circle,S5line.Assume that particleA is on the blade edge of the front chamber. To establish an initial coordinate system XYZ, the Mars surface can be set to plane XOY, and the origin O is set to the initial position of particle A. Because the actual movement trajectory parameters are undocumented, the trajectory calculation here is based on the assuming parameters as follows. For the whole sampling process, assume that the linear speed of the chamber remains the same at 5 mm/s, and the distance between particle A and rotation axis is 100 mm. For stage one, the tilting angle, which is defined as the angle between the digging direction and the surface, is set to 30°, which is assumed to provide the maximum insertion force. For stage three, the sampling depth is set to 10 cm, and the maximum horizontal displacement of RA is set to 500 mm. For stage four, to prevent sampling loss during the lifting process, the minimum tilting angle for the front chamber is set to 75°.According to the auxiliary coordinate system shown in Figure4, to simplify the calculation, the plane Au can be set to a vertical plane parallel to plane XOZ. Hence, the positional parameters are β=0 and θ=π/2 (in the computer calculation, the angle values must be in radians); thus, the input parameter matrix MSP for this assuming movement trajectory can be obtained, as given by the following:
(18)MSP=34.640−π/60π/2510.47100−π/60π/2539.30000π/2526.1810000π/255.3605π/120π/25.Introducing the matrixMSP into the program described in Figures 7 and 8, the assumed movement trajectory of particle A on the front chamber can be calculated, as shown by the solid line in Figure 10. If the width of the front chamber is 15 mm, then the trajectory of particle A can be moved to both sides by 7.5 cm to obtain the digging surface, as shown by the dashed line in Figure 10.Figure 10
Movement trajectory calculation model.


## 3. Structure Module Group

### 3.1. Structure Module Definition

The main target for the new design model is to analyse the interaction relationship between the sampling devices and the samples. According to this target, the structure modules discussed in this paper are focused on those structures that will contact with the sample and generate relative movement. For those structures that do not contact the sample, they hardly affect the analysis of interaction relationship and thus are not considered in the following discussion.Through the investigation of the planetary sampling equipment in the introduction chapters, for deep sampling, the sampling structures around the world nearly have the same external shape, which is typically a spiral. In addition, the only distinction is in the specific parameter values, such as diameter, length, and helix angle. In terms of surface sampling, because of the different sampling requirements and space, there are obvious different structure designs shown in Figure11. In addition, these obvious differences bring about significant difficulty on the analysis of interaction relationship. For example, when designing a piece of sampling equipment, traditionally, the structure that comes into contact with the sample is to be evaluated as a whole; i.e., the calculation, simulation, and experiments for this structure are disposable. If there is any microscopic modification or optimization of this structure, all of the studies must be repeated, leading to substantial waste of both research resources and time.Figure 11
Planetary surface sampling devices around the world. (a–c) surface sampling equipment used in the Apollo Projects [17]; (d) SRR2k designed for the Mars 2018 project [18]; (e) digging equipment designed by Honeybee Corp. [19]; (f) sampling equipment designed by Keio University [20]; (g–i) surface sampling equipment designed for the Chinese lunar project [21–24].
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)To solve this problem, establishing a relatively unified and flexible structure model is necessary, as this model can transform smoothly among different structures without repeating any of the research efforts for potential modification or optimization. Modular design for the sampling structure is the key to this solution. Essentially, every surface sampling structure can be viewed as consisting of multiple regular modules. Because of the regular external shapes, the calculation, simulation, or experiment works of these modules can easily achieve results with higher accuracy. More importantly, these results can be reused. For example, for a random sampling structure that consists of several modules, the analysis result of this structure can be achieved by combining the results of the analysis of the selected modules. When the structure must be modified, instead of repeating the calculation, simulation, and experiment works, the design work can be simplified to add or remove several modules via their analysis results. As shown in Figure11, although the sampling devices are quite different around the world, each of them can be viewed as being composed of several flat plates or curved plates. Hence, the flat plate (curved plate) is selected as the basic module of the structure module group.


### 3.2. Basic Modules in the Structure Group

To meet the requirements of modular design and bring convenience to module combination, the structure modules should share a set of unified parameters. Traditionally, the parametric definition for any equipment is focused on its integral external structure. However, the parameter types (or numbers) are typically quite different among the various structures; as a result, this type of parametric definition cannot be used for the structure module. The parameters for structure modules here are defined according to their manufacturing processes:(1)
Section parameters. As shown in Figure12(a), the cross section of the flat plate is rectangular and that for the curved plate can be considered as a bent rectangle. Therefore, the section parameters contain the rectangle length LL, rectangle width LW and bending radius RL. The bending radius is RL=0 for flat plates. For curved plates, the rectangle length LL denotes the length of the maximum arc, and the bending radius RL is the radius of the maximum arc
(2)
Extension parameters. After establishing the cross section, define the extension lengthLE to enable it to extend toward the vertical direction
(3)
Cutting parameters. Cutting will dramatically improve the flexibility and variability of the flat plate (curved plate). The cutting here is similar to the scan-removal operation in the tridimensional CAD software. As shown in Figure12(c), the shaded areas will be cut and removed. First, the initial position of cutting is defined to be on the left edge of the upper surface along the extension direction, and the distance between the initial position and cross section is LCLC∈0,LE. Next, define the cutting contour on horizontal planes as well as the scan path on the vertical planes, and have the cutting contour perform “scan-removal” work along the scan path. According to the derivation in Section 2.2, any graphic is comprised of a segment and an arc, both of which can be fully defined by the initial speed, angle, and radius (the value of the radius for segments is equal to zero). For the cutting parameters, speed is completely unnecessary, whereas the angle and radius are required for both the cutting contour and the scan path. The initial angle of the cutting contour is defined as γHC, where γHC∈0,180, and its positive direction is clockwise from the extension direction of the cross section. In addition, the initial angle of the scan path is γVC, where γVC∈0,180, and its positive direction is counterclockwise from the extension direction of cross section. Finally, the initial radius for the cutting contour and the scan path are defined as RHC and RVCFigure 12
Definitions of the basic structure module.
(a)
(b)
(c)Thus, the mathematical definition for basic structure moduleT is
(19)T=TLL,Lw,RL,LE,LC,γHC,RHC,γVC,RVC.According to the modular design theory, a sampling device consists of several basic modules; assuming the number ism, this sampling device can be defined by matrix MT:
(20)MT=LL1LW1RL1LE1LC1γHC1RHC1γVC1RVC1LL2LW2RL2LE2LC2γHC2RHC2γVC2RVC2LL3LW3RL3LE3LC3γHC3RHC3γVC3RVC3⋮⋮⋮⋮⋮⋮⋮⋮⋮LLmLWmRLmLEmLCmγHCmRHCmγVCmRVCm.


### 3.3. Application of the Structure Group

Equation (19) defines the basic modules of the structure group by using a set of unified parameters, and equation (20) defines random surface sampling devices using a parameter matrix. For example, the digging spoon used in the Apollo Projects, as shown in Figure 11(c), can be decomposed into nine basic modules (Figure 13). In these modules, modules 1, 3, 5, 6, and 8 are flat plates and modules 2, 4, 7, and 9 are curved plates. The parameter definition of these basic modules is shown in Table 1.Figure 13
Disassembling schematic of the digging spoon used in the Apollo missions.
(a)
(b)Table 1
Parameter definition of the digging spoon.
Module no.
Cross section
Cutting contour
Scan path
Zero value parameters
Lateral deflection
1
Flat
Arc
Segment
RL1,RVC1
2, 9
Curved
Segment
Segment
RHC2,RVC2
90°
3, 8
Flat
Segment
Segment
RL3,RHC3,RVC3
90°
4, 7
Curved
Segment
Segment
RHC4,RVC4
90°
5
Flat
Arc
Segment∗
RL5,RVC5
6∗∗
Flat
Arc
Segment
RL6,RVC6
∗The scan path for module 5 is nonvertical; ∗∗the original shape for module 6 is a flat plate, where four corners should perform arc cutting, considering this plate is of symmetrical shape with two symmetrical axes; hence, the plate can be split into four identical plates along the vertical and horizontal axis.Assuming that the thickness of the digging spoon is 5 mm, combined with the relative proportion of its structure, the assumed defining matrixMT for the digging spoon is
(21)MT=T1T2T3T4T5T6∗T7T8T9=12050250229.652.598.590062.8520217.419232.109004050192128.232.1090062.8520128.295.932.1090012050103.9103.9135.484.532.1040508060902090062.8520128.2128.2122.109004050192192122.1090062.8520217.4217.4122.10900.where ∗ means there are four identical modules T6 with identical parameters; here, only one of them is listed.


## 3.1. Structure Module Definition

The main target for the new design model is to analyse the interaction relationship between the sampling devices and the samples. According to this target, the structure modules discussed in this paper are focused on those structures that will contact with the sample and generate relative movement. For those structures that do not contact the sample, they hardly affect the analysis of interaction relationship and thus are not considered in the following discussion.Through the investigation of the planetary sampling equipment in the introduction chapters, for deep sampling, the sampling structures around the world nearly have the same external shape, which is typically a spiral. In addition, the only distinction is in the specific parameter values, such as diameter, length, and helix angle. In terms of surface sampling, because of the different sampling requirements and space, there are obvious different structure designs shown in Figure11. In addition, these obvious differences bring about significant difficulty on the analysis of interaction relationship. For example, when designing a piece of sampling equipment, traditionally, the structure that comes into contact with the sample is to be evaluated as a whole; i.e., the calculation, simulation, and experiments for this structure are disposable. If there is any microscopic modification or optimization of this structure, all of the studies must be repeated, leading to substantial waste of both research resources and time.Figure 11
Planetary surface sampling devices around the world. (a–c) surface sampling equipment used in the Apollo Projects [17]; (d) SRR2k designed for the Mars 2018 project [18]; (e) digging equipment designed by Honeybee Corp. [19]; (f) sampling equipment designed by Keio University [20]; (g–i) surface sampling equipment designed for the Chinese lunar project [21–24].
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)To solve this problem, establishing a relatively unified and flexible structure model is necessary, as this model can transform smoothly among different structures without repeating any of the research efforts for potential modification or optimization. Modular design for the sampling structure is the key to this solution. Essentially, every surface sampling structure can be viewed as consisting of multiple regular modules. Because of the regular external shapes, the calculation, simulation, or experiment works of these modules can easily achieve results with higher accuracy. More importantly, these results can be reused. For example, for a random sampling structure that consists of several modules, the analysis result of this structure can be achieved by combining the results of the analysis of the selected modules. When the structure must be modified, instead of repeating the calculation, simulation, and experiment works, the design work can be simplified to add or remove several modules via their analysis results. As shown in Figure11, although the sampling devices are quite different around the world, each of them can be viewed as being composed of several flat plates or curved plates. Hence, the flat plate (curved plate) is selected as the basic module of the structure module group.


## 3.2. Basic Modules in the Structure Group

To meet the requirements of modular design and bring convenience to module combination, the structure modules should share a set of unified parameters. Traditionally, the parametric definition for any equipment is focused on its integral external structure. However, the parameter types (or numbers) are typically quite different among the various structures; as a result, this type of parametric definition cannot be used for the structure module. The parameters for structure modules here are defined according to their manufacturing processes:(1)
Section parameters. As shown in Figure12(a), the cross section of the flat plate is rectangular and that for the curved plate can be considered as a bent rectangle. Therefore, the section parameters contain the rectangle length LL, rectangle width LW and bending radius RL. The bending radius is RL=0 for flat plates. For curved plates, the rectangle length LL denotes the length of the maximum arc, and the bending radius RL is the radius of the maximum arc
(2)
Extension parameters. After establishing the cross section, define the extension lengthLE to enable it to extend toward the vertical direction
(3)
Cutting parameters. Cutting will dramatically improve the flexibility and variability of the flat plate (curved plate). The cutting here is similar to the scan-removal operation in the tridimensional CAD software. As shown in Figure12(c), the shaded areas will be cut and removed. First, the initial position of cutting is defined to be on the left edge of the upper surface along the extension direction, and the distance between the initial position and cross section is LCLC∈0,LE. Next, define the cutting contour on horizontal planes as well as the scan path on the vertical planes, and have the cutting contour perform “scan-removal” work along the scan path. According to the derivation in Section 2.2, any graphic is comprised of a segment and an arc, both of which can be fully defined by the initial speed, angle, and radius (the value of the radius for segments is equal to zero). For the cutting parameters, speed is completely unnecessary, whereas the angle and radius are required for both the cutting contour and the scan path. The initial angle of the cutting contour is defined as γHC, where γHC∈0,180, and its positive direction is clockwise from the extension direction of the cross section. In addition, the initial angle of the scan path is γVC, where γVC∈0,180, and its positive direction is counterclockwise from the extension direction of cross section. Finally, the initial radius for the cutting contour and the scan path are defined as RHC and RVCFigure 12
Definitions of the basic structure module.
(a)
(b)
(c)Thus, the mathematical definition for basic structure moduleT is
(19)T=TLL,Lw,RL,LE,LC,γHC,RHC,γVC,RVC.According to the modular design theory, a sampling device consists of several basic modules; assuming the number ism, this sampling device can be defined by matrix MT:
(20)MT=LL1LW1RL1LE1LC1γHC1RHC1γVC1RVC1LL2LW2RL2LE2LC2γHC2RHC2γVC2RVC2LL3LW3RL3LE3LC3γHC3RHC3γVC3RVC3⋮⋮⋮⋮⋮⋮⋮⋮⋮LLmLWmRLmLEmLCmγHCmRHCmγVCmRVCm.


## 3.3. Application of the Structure Group

Equation (19) defines the basic modules of the structure group by using a set of unified parameters, and equation (20) defines random surface sampling devices using a parameter matrix. For example, the digging spoon used in the Apollo Projects, as shown in Figure 11(c), can be decomposed into nine basic modules (Figure 13). In these modules, modules 1, 3, 5, 6, and 8 are flat plates and modules 2, 4, 7, and 9 are curved plates. The parameter definition of these basic modules is shown in Table 1.Figure 13
Disassembling schematic of the digging spoon used in the Apollo missions.
(a)
(b)Table 1
Parameter definition of the digging spoon.
Module no.
Cross section
Cutting contour
Scan path
Zero value parameters
Lateral deflection
1
Flat
Arc
Segment
RL1,RVC1
2, 9
Curved
Segment
Segment
RHC2,RVC2
90°
3, 8
Flat
Segment
Segment
RL3,RHC3,RVC3
90°
4, 7
Curved
Segment
Segment
RHC4,RVC4
90°
5
Flat
Arc
Segment∗
RL5,RVC5
6∗∗
Flat
Arc
Segment
RL6,RVC6
∗The scan path for module 5 is nonvertical; ∗∗the original shape for module 6 is a flat plate, where four corners should perform arc cutting, considering this plate is of symmetrical shape with two symmetrical axes; hence, the plate can be split into four identical plates along the vertical and horizontal axis.Assuming that the thickness of the digging spoon is 5 mm, combined with the relative proportion of its structure, the assumed defining matrixMT for the digging spoon is
(21)MT=T1T2T3T4T5T6∗T7T8T9=12050250229.652.598.590062.8520217.419232.109004050192128.232.1090062.8520128.295.932.1090012050103.9103.9135.484.532.1040508060902090062.8520128.2128.2122.109004050192192122.1090062.8520217.4217.4122.10900.where ∗ means there are four identical modules T6 with identical parameters; here, only one of them is listed.


## 4. Coupling Analysis between the Motion and Structure Groups

### 4.1. Principle of Coupling Analysis

Coupling analysis describes the connection between the motion and structure groups, and the coupling also plays an important role in achieving accurate research results. As shown in Figure14(a), the entire sampling trajectory can be divided into several basic motion modules; similarly, the entire sampling equipment is composed of several basic structure modules. Hence, the whole sampling process transforms into two mathematical aggregations, and the coupling analysis can be simplified to study the connection between the basic motion module and the basic structure module.Figure 14
Principle of coupling analysis for the motion and structure groups.
(a)
(b)
(c)
(d)As shown in Figures14(b)–14(d), there are only three types of applications in the connection studies:
(1)
Analysis of one structure module connected with all movement trajectories to obtain the force analysis for one specific module in the whole sampling process (Figure14(b))
(2)
Analysis of all structure modules connected with one movement trajectory to obtain the force analysis of the entire sampling device in one specific sampling process (Figure14(c))
(3)
Analysis of all structure modules connected with all movement trajectories to obtain the force analysis of the entire sampling device in the whole sampling process (Figure14(d))The core of the connection research is to analyse the interaction relationship between the single motion module and the single structure module at any given moment. In the motion groups, tridimensional uniform rectilinear motion and tridimensional uniform circular motion are the basic modules. The major difference between the two modules is the velocity during the entire movement process. For rectilinear motion, both the initial value and direction remain the same, whereas for circular motion, only the initial value will not change, and its direction changes with time. Therefore, in the connection research, the primary function for basic motion modules is to calculate the instantaneous velocity value and direction at any given moment.In the structure groups, the flat and curved plates are the two basic modules. To obtain the entire force distribution of a basic module in the sampling process, force analysis must be performed on every surface of this module, according to the contact relationship between the surfaces and samples. First, the velocity direction for each surface should be obtained through the calculation of the basic motion modules. Next, the velocity direction is split into two directions: parallel to the surface and perpendicular to the surface. According to the definition of the basic structure module, there are only two types of surfaces: flat and curved surfaces. Two types of velocity directions combined with two types of surfaces result in four different coupling analyses, as shown in Figure15.Figure 15
Coupling force analysis between the velocity and the surfaces.When the velocity direction is parallel to the surface (Figures15(a) and 15(b)), the friction, which is the only force acting on the surface, is spread uniformly on the surface. When the velocity direction is perpendicular to the surface (Figures 15(c) and 15(d)), the only force acting on the surface is the reaction, which is also spread uniformly on the surface. The direction of both forces is opposite to the velocity direction. To calculate the resultant force for a flat surface, the calculation method is the surface area multiplied by the force per unit area. For a curved surface, the first step is to flatten it as a rectangular plane and then calculate the parameters as a flat surface.By combining the force from every surface in every direction, the force analysis of a specific structure module at a given moment can be calculated. Similarly, the force analysis for a given sampling device can be calculated by combining the force from every structure module, which completes the connection study, i.e., the coupling analysis.


### 4.2. Examples of Coupling Analysis

Here, two examples are provided to briefly show the coupling analysis based on the motion and structure module groups. Assume that the digging spoon shown in Figure13 is used to perform sampling; its movement trajectory is shown in Figure 10. Use the bottom structure module T1 to perform coupling analysis in movement stage one S1line and stage two S2circle.


#### 4.2.1. Coupling Analysis for Structure ModuleT1 in Movement Stage S1line

From equation (21), the parameter value of structure module T1 is
(22)T1=T120,5,0,250,229.6,52.5,98.5,90,0.Assume that the planetary surface is the planeX=0,Y=0, and the origin of coordinate system is the first point where the structure makes contact with the surface. The movement equation S1line of structure module T1 in stage one can be obtained by using equation (18):
(23)S1line=Sline0,0,0,5,−π/6,0,π/2,34.64,0.As shown in Figure16, the solid line represents the trajectory S1line, and the structure module T1 will perform a sampling movement along the trajectory. To enhance the display effect, in Figure 16, the thickness of structure module T1 is increased; however, in actual calculation, the thickness should retain the origin value. Based on the principle of coupling analysis in Section 4.1, to analyse the entire force of structure module T1, the primary task is to distinguish the contact areas because the contact generates forces and contact areas are varied among different surfaces. For structure module T1, the bottom surface, upper surface, and left and right surfaces are parallel to the direction of velocity; therefore, the forces on these surfaces are friction, denoted as Fu1,Fd1,Fl1,Fr1, respectively. These frictional forces are caused by stress induced by depth and sample gravity in movement. Similarly, the front surface of structure module T1 is a curved surface perpendicular to the velocity direction, which brings the reaction Ft1 from the sample cutting resistance and bearing capacity corresponding to the movement. Hence, the total force for structure module T1 in stage S1line is
(24)FT1S1=Fu1+Fd1+Fl1+Fr1+Ft1.Figure 16
Force analysis for structure moduleT1 in stage S1line.


#### 4.2.2. Coupling Analysis for Structure ModuleT1 in Movement Stage S2circle

Assuming that the end position of stageS1line is Pt1, the movement equation of structure module T1 in stage S2circle is given by the following:
(25)S2circle=ScirclePt1,5,−π/6,0,π/2,10.47,100.When structure moduleT1 performs a sampling work in stage S2circle, the trajectory is an arc (solid line in Figure 17). During the whole stage, the velocity direction is neither parallel nor perpendicular to the midaxis of structure module T1 at any moment. Therefore, in the force analysis, the velocity v must be split into the component velocity vh parallel to the midaxis and the component velocity vv perpendicular to the midaxis.Figure 17
The force analysis for structure moduleT1 in stage S2circle.Under the influence of velocityvh, similar to stage S1line, the friction is spread uniformly on the bottom surface, upper surface, and left and right surfaces, represented as Fu2h,Fd2h,Fl2h, and Fr2h respectively. The reaction Ft2h is perpendicular to the front surface of structure module T1.Under the influence of velocityvv, the friction, denoted as Fl2v,Fr2v,Ft2v, is spread throughout the front surface, left surface, and right surface. These frictional forces also originate from the stress caused by the depth and sample gravity during movement. For the upper surface, there is the reaction Fu2v induced by sample gravity. For the bottom surface, no force will be acting on it because this surface will detach from the sample.Hence, the total force for structure moduleT1 in stage S2circle is
(26)FT1S2_h=Fu2h+Fd2h+Fl2h+Fr2h+Ft2h,FT1S2_v=Fu2v+Fl2v+Fr2v+Ft2v.


## 4.1. Principle of Coupling Analysis

Coupling analysis describes the connection between the motion and structure groups, and the coupling also plays an important role in achieving accurate research results. As shown in Figure14(a), the entire sampling trajectory can be divided into several basic motion modules; similarly, the entire sampling equipment is composed of several basic structure modules. Hence, the whole sampling process transforms into two mathematical aggregations, and the coupling analysis can be simplified to study the connection between the basic motion module and the basic structure module.Figure 14
Principle of coupling analysis for the motion and structure groups.
(a)
(b)
(c)
(d)As shown in Figures14(b)–14(d), there are only three types of applications in the connection studies:
(1)
Analysis of one structure module connected with all movement trajectories to obtain the force analysis for one specific module in the whole sampling process (Figure14(b))
(2)
Analysis of all structure modules connected with one movement trajectory to obtain the force analysis of the entire sampling device in one specific sampling process (Figure14(c))
(3)
Analysis of all structure modules connected with all movement trajectories to obtain the force analysis of the entire sampling device in the whole sampling process (Figure14(d))The core of the connection research is to analyse the interaction relationship between the single motion module and the single structure module at any given moment. In the motion groups, tridimensional uniform rectilinear motion and tridimensional uniform circular motion are the basic modules. The major difference between the two modules is the velocity during the entire movement process. For rectilinear motion, both the initial value and direction remain the same, whereas for circular motion, only the initial value will not change, and its direction changes with time. Therefore, in the connection research, the primary function for basic motion modules is to calculate the instantaneous velocity value and direction at any given moment.In the structure groups, the flat and curved plates are the two basic modules. To obtain the entire force distribution of a basic module in the sampling process, force analysis must be performed on every surface of this module, according to the contact relationship between the surfaces and samples. First, the velocity direction for each surface should be obtained through the calculation of the basic motion modules. Next, the velocity direction is split into two directions: parallel to the surface and perpendicular to the surface. According to the definition of the basic structure module, there are only two types of surfaces: flat and curved surfaces. Two types of velocity directions combined with two types of surfaces result in four different coupling analyses, as shown in Figure15.Figure 15
Coupling force analysis between the velocity and the surfaces.When the velocity direction is parallel to the surface (Figures15(a) and 15(b)), the friction, which is the only force acting on the surface, is spread uniformly on the surface. When the velocity direction is perpendicular to the surface (Figures 15(c) and 15(d)), the only force acting on the surface is the reaction, which is also spread uniformly on the surface. The direction of both forces is opposite to the velocity direction. To calculate the resultant force for a flat surface, the calculation method is the surface area multiplied by the force per unit area. For a curved surface, the first step is to flatten it as a rectangular plane and then calculate the parameters as a flat surface.By combining the force from every surface in every direction, the force analysis of a specific structure module at a given moment can be calculated. Similarly, the force analysis for a given sampling device can be calculated by combining the force from every structure module, which completes the connection study, i.e., the coupling analysis.


## 4.2. Examples of Coupling Analysis

Here, two examples are provided to briefly show the coupling analysis based on the motion and structure module groups. Assume that the digging spoon shown in Figure13 is used to perform sampling; its movement trajectory is shown in Figure 10. Use the bottom structure module T1 to perform coupling analysis in movement stage one S1line and stage two S2circle.


### 4.2.1. Coupling Analysis for Structure ModuleT1 in Movement Stage S1line

From equation (21), the parameter value of structure module T1 is
(22)T1=T120,5,0,250,229.6,52.5,98.5,90,0.Assume that the planetary surface is the planeX=0,Y=0, and the origin of coordinate system is the first point where the structure makes contact with the surface. The movement equation S1line of structure module T1 in stage one can be obtained by using equation (18):
(23)S1line=Sline0,0,0,5,−π/6,0,π/2,34.64,0.As shown in Figure16, the solid line represents the trajectory S1line, and the structure module T1 will perform a sampling movement along the trajectory. To enhance the display effect, in Figure 16, the thickness of structure module T1 is increased; however, in actual calculation, the thickness should retain the origin value. Based on the principle of coupling analysis in Section 4.1, to analyse the entire force of structure module T1, the primary task is to distinguish the contact areas because the contact generates forces and contact areas are varied among different surfaces. For structure module T1, the bottom surface, upper surface, and left and right surfaces are parallel to the direction of velocity; therefore, the forces on these surfaces are friction, denoted as Fu1,Fd1,Fl1,Fr1, respectively. These frictional forces are caused by stress induced by depth and sample gravity in movement. Similarly, the front surface of structure module T1 is a curved surface perpendicular to the velocity direction, which brings the reaction Ft1 from the sample cutting resistance and bearing capacity corresponding to the movement. Hence, the total force for structure module T1 in stage S1line is
(24)FT1S1=Fu1+Fd1+Fl1+Fr1+Ft1.Figure 16
Force analysis for structure moduleT1 in stage S1line.


### 4.2.2. Coupling Analysis for Structure ModuleT1 in Movement Stage S2circle

Assuming that the end position of stageS1line is Pt1, the movement equation of structure module T1 in stage S2circle is given by the following:
(25)S2circle=ScirclePt1,5,−π/6,0,π/2,10.47,100.When structure moduleT1 performs a sampling work in stage S2circle, the trajectory is an arc (solid line in Figure 17). During the whole stage, the velocity direction is neither parallel nor perpendicular to the midaxis of structure module T1 at any moment. Therefore, in the force analysis, the velocity v must be split into the component velocity vh parallel to the midaxis and the component velocity vv perpendicular to the midaxis.Figure 17
The force analysis for structure moduleT1 in stage S2circle.Under the influence of velocityvh, similar to stage S1line, the friction is spread uniformly on the bottom surface, upper surface, and left and right surfaces, represented as Fu2h,Fd2h,Fl2h, and Fr2h respectively. The reaction Ft2h is perpendicular to the front surface of structure module T1.Under the influence of velocityvv, the friction, denoted as Fl2v,Fr2v,Ft2v, is spread throughout the front surface, left surface, and right surface. These frictional forces also originate from the stress caused by the depth and sample gravity during movement. For the upper surface, there is the reaction Fu2v induced by sample gravity. For the bottom surface, no force will be acting on it because this surface will detach from the sample.Hence, the total force for structure moduleT1 in stage S2circle is
(26)FT1S2_h=Fu2h+Fd2h+Fl2h+Fr2h+Ft2h,FT1S2_v=Fu2v+Fl2v+Fr2v+Ft2v.


## 4.2.1. Coupling Analysis for Structure ModuleT1 in Movement Stage S1line

From equation (21), the parameter value of structure module T1 is
(22)T1=T120,5,0,250,229.6,52.5,98.5,90,0.Assume that the planetary surface is the planeX=0,Y=0, and the origin of coordinate system is the first point where the structure makes contact with the surface. The movement equation S1line of structure module T1 in stage one can be obtained by using equation (18):
(23)S1line=Sline0,0,0,5,−π/6,0,π/2,34.64,0.As shown in Figure16, the solid line represents the trajectory S1line, and the structure module T1 will perform a sampling movement along the trajectory. To enhance the display effect, in Figure 16, the thickness of structure module T1 is increased; however, in actual calculation, the thickness should retain the origin value. Based on the principle of coupling analysis in Section 4.1, to analyse the entire force of structure module T1, the primary task is to distinguish the contact areas because the contact generates forces and contact areas are varied among different surfaces. For structure module T1, the bottom surface, upper surface, and left and right surfaces are parallel to the direction of velocity; therefore, the forces on these surfaces are friction, denoted as Fu1,Fd1,Fl1,Fr1, respectively. These frictional forces are caused by stress induced by depth and sample gravity in movement. Similarly, the front surface of structure module T1 is a curved surface perpendicular to the velocity direction, which brings the reaction Ft1 from the sample cutting resistance and bearing capacity corresponding to the movement. Hence, the total force for structure module T1 in stage S1line is
(24)FT1S1=Fu1+Fd1+Fl1+Fr1+Ft1.Figure 16
Force analysis for structure moduleT1 in stage S1line.


## 4.2.2. Coupling Analysis for Structure ModuleT1 in Movement Stage S2circle

Assuming that the end position of stageS1line is Pt1, the movement equation of structure module T1 in stage S2circle is given by the following:
(25)S2circle=ScirclePt1,5,−π/6,0,π/2,10.47,100.When structure moduleT1 performs a sampling work in stage S2circle, the trajectory is an arc (solid line in Figure 17). During the whole stage, the velocity direction is neither parallel nor perpendicular to the midaxis of structure module T1 at any moment. Therefore, in the force analysis, the velocity v must be split into the component velocity vh parallel to the midaxis and the component velocity vv perpendicular to the midaxis.Figure 17
The force analysis for structure moduleT1 in stage S2circle.Under the influence of velocityvh, similar to stage S1line, the friction is spread uniformly on the bottom surface, upper surface, and left and right surfaces, represented as Fu2h,Fd2h,Fl2h, and Fr2h respectively. The reaction Ft2h is perpendicular to the front surface of structure module T1.Under the influence of velocityvv, the friction, denoted as Fl2v,Fr2v,Ft2v, is spread throughout the front surface, left surface, and right surface. These frictional forces also originate from the stress caused by the depth and sample gravity during movement. For the upper surface, there is the reaction Fu2v induced by sample gravity. For the bottom surface, no force will be acting on it because this surface will detach from the sample.Hence, the total force for structure moduleT1 in stage S2circle is
(26)FT1S2_h=Fu2h+Fd2h+Fl2h+Fr2h+Ft2h,FT1S2_v=Fu2v+Fl2v+Fr2v+Ft2v.


## 5. Conclusion

A new modular motion-structure design model for surface sampling that freely combines the basic modules from the motion and structure groups was discussed in this paper. This model can be applied to different movements and equipment structures for more accurate definition and analysis results as follows:(1)
The main function of the motion group is to analyse the sampling movement trajectory and to obtain the required coordinates at any given moment during the sampling process. There are two basic modules in this group: tridimensional uniform rectilinear motion and tridimensional uniform circular motion. With the same input and output parameters, the two types of basic motion modules can be used to compose any sampling trajectory, as needed
(2)
To adapt to the various external structures of the surface sampling equipment, the flat plate and the curved plate are selected as the basic modules in the structure group. Sharing the same parameters (including section, extension, and cutting parameters), the two types of basic modules can change their external shape by adjusting their parameters. More importantly, the same parameters make it convenient to combine different structure basic modules to define a complex sampling structure
(3)
Based on the motion group and the structure group, the research on the entire sampling process transforms into the research studies on the motion groups, the structure groups and the coupling analysis between the two groups. The different types of combinations among the motion modules and the structure modules will lead to different results from the coupling analysis. According to the characteristics of the two groups, the key to performing the coupling analysis is to determine the force on the flat or curved surface as well as the influence of the velocity direction



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*Source: 5987306-2019-04-01.xml*