# Collision Sensing Using Force/Torque Sensor

Authors: Yu-Quan Leng; Zheng-Cang Chen; Xu He; Yang Zhang; Wei Zhang
Journal: Journal of Sensors
 (2016)

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## Abstract

Collision sensing including collision position, collision direction, and force size could make robots smoothly interact with environment, so that the robots can strongly adapt to the outside world. Skin sensor imitates principles of human skin using special material and physical structure to obtain collision information, but this method has some disadvantages, such as complex design, low sampling rate, and poor generality. In this paper, a new method using force/torque sensor to calculate collision position, collision direction, and force size is proposed. Detailed algorithm is elaborated based on physical principle and unified modeling method for basic geometric surface. Gravity compensation and dynamic compensation are also introduced for working manipulators/robots in gravity and dynamic environment. In addition, considering algorithm solvability and uniqueness, four constraints are proposed, which are force constraint, geometric constraint, normal vector constraint, and current mutation constraint. In order to solve conflict solution of algorithm in redundant constraints, compatibility solution analysis is proposed. Finally, a simulation experiment shows that the proposed method can achieve collision information efficiently and accurately.


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## Body

## 1. Introduction

In recent years, more and more sensors are installed on robots to realize sensing and interacting with environment. These sensors help robots achieve a large amount of information about the environment, which contributes robots to make correct decision and planning. For example, vision sensor assists robots in obtaining appearance information of environment, which is used to realize target tracking and object capture, and so forth.Collision sensing is also an important environment sensing method, through which robots sense collision point and collision direction to avoid collision again, and makes compliment control come true [1, 2]. In the environment containing human and robots, collision sensing is extremely significant to avoid serious collision. In addition, in [3], collision sensing is also used for compliance assembly. Since that collision sensing has so many applications, realization of whole robotic body sensing is gradually carried out [4, 5].Skin force sensor like human skin system, which is pasted on the whole surface of robots, could sense force action point, force direction and size [2, 5]. In the research filed, MIT has developed a special compound material called QTC (Quantum Tunneling Composite), which is pressed by metal and nonmetal [2, 6]. This sensor measures force information by resistance change when it is pressed. Similar researches also include organic thin film composed of pentacene molecular (U-Tokyo, Japan) and E-skin (Philips Company), and so forth [1]. The skin force sensor looks like an ideal solution for robot force sensing, but there are many disadvantages that make it hard to be used in real world, such as being easy to damage during collision, expensive price, complex design and high environment requirements, low sampling rate, and poor generality [4, 7]. Moreover, sensitivity of skin sensor system could not achieve information of collision direction and could be influenced by many factors, such as temperature and humidity [1].Skin force sensor is the direct method to measure force information; on the other way, it also could be checked out through indirect techniques [8, 9]. In [10] Mason and Salisbury have proposed the theory of calculating force information using force/torque sensor and in [11] Bicchi et al. have developed experiments to verify it. Zhou and Han use the theory to realize skin force sense on the one freedom finger [12–14]. In this paper, theories of manipulator with multiple degrees of freedom are applied and method of compatibility solution analysis to solve contradiction of solutions is proposed.The rest of this paper is organized as follows. Section2 describes detection principle, when point collision happens, unified equation for collision sensing considering gravity and manipulator dynamics, and unified modeling method for basic geometric surface. In the gravity environment, force analysis is proposed in Section 3. In Section 4, force analysis is shown when manipulator is in a working condition. Solvability analysis is provided in Section 5, which includes force constraint, geometric constraint, normal vector constraint, current mutation constraint, and compatibility solution analysis. In Section 6, the simulation experiment and results are shown to prove effectiveness of the algorithm. Finally, the concluding remarks follow in Section 7.


## 2. Detection Principle

In this section, physical principle about point collision on manipulator is introduced. On the one hand, the whole process of force effect needs meets force analysis principle, and on the other hand, the result of calculation should suit physical laws, such as pointc which is on the manipulator surface and the angle of force direction and normal vector is less than π/2.


### 2.1. Force Analysis

Assume that no gravity and manipulator are static; force analysis is shown in Figure1. In order to apperceive whole system, force/torque sensor is assembled on the base and close to the first joint. The force/torque sensor could measure force in three directions and torque rotated of three coordinate axes.Figure 1
Collision analysis.In Figure1, point C means collision point, which is assumed on ith joint. The center of force/torque sensor is expressed as B, which is set as origin of B coordinate system. Every joint of manipulator is built coordinate system as Ti according to DH parameter method. Ti represents coordinate system of ith joint, and p, f, and τ, respectively, mean collision force, measured force, and measured torque. nc represents normal vector of point c. Tc could be achieved by translating coordinate system Ti and no rotation. Then coordinate of point C in TB is obtained by coordinate transformation, as shown in the following:(1)TBc=TB1T12⋯Ti-1iTic,where(2)Tii-1=cθi-sθicαisθisαiaicθisθicθicαi-cθisαiaisθi0sαicαidi0001,Tic=100xic010yic001zic0001.When collision occurs, the outside force isp=(fcx,fcy,fcz)T, and data of force/torque sensor are f=(fx,fy,fz)T and τ=(τx,τy,τz)T. According to force analysis, data of outside force in TB should be equal to measured data in ideal condition, as shown in the following:(3)f=fcB,τ=τcB,where(4)fcB=fcBx,fcBy,fcBzT=Tp-TTBc3,1T,(5)τcB=τcBx,τcBy,τcBzT=cB×p=Tc-iTBc3,1×p.In (4),(6)p-=fcx,fcy,fcz,1Tand T(·)3,1 means extraction of elements of the first three rows and the first column.In (5),(7)c-i=xci,yci,zci,1T.Considering factor of gravity and manipulator dynamics in real world, general equation could be shown as follows:(8)f=fcB+fGB+fdBτ=τcB+τGB+τdB.In (8), fcB and τcB, respectively, express force and torque caused by collision mapping in basic coordinate system B; fGB and τGB, respectively, mean force and torque caused by gravity mapping in basic coordinate system B, which will be analyzed in Section 3; fdB and τdB, respectively, express force and torque caused by manipulator dynamics mapping in basic coordinate system B, which will be shown in Section 4.In (8), fGB, τGB, fdB, and τdB could be calculated based on states of manipulator and joint physical properties, such as joint angle, joint velocity, joint acceleration, and joint quality. Collision information is just contented in fcB and τcB, that is, the parameters of fcx, fcy, fcz, xci, yci, and zci.


### 2.2. Physical Constraint

Normal geometrical surface contains ellipsoid, sphere, cylindrical surface, and plane. They are able to construct all simple surfaces including outline of manipulators. Mathematical description mode of all manipulator surface is achieved through coordinate transformation. From the role of geometrical change, the core of four types of surface is ellipsoid, which is able to change to other three types by changing parameters, as shown in Figure2.Figure 2
Unified modeling method for basic geometric surface.Assume three axis lengths of ellipsoid are2α, 2β, and 2γ, setting center as origin of coordinate system. When α=β=γ, sphere will be formed; when γ→+∞, cylindrical surface is achieved; and when α→+∞, β→0, and γ→∞, plane surface is displayed. In summary, it just needs to build ellipsoid mode to describe all surface, as shown in the following:(9)Sc=cTATAc=1,where(10)A=1α0001β0001γ.Normal vector of pointC is set as nc and could be calculated with equation as follows:(11)nc=∇Sc∇Sc=2A2c2A2c=A2cA2c,where c=(x,y,z) means coordinate values and ∇ represents gradient.


## 2.1. Force Analysis

Assume that no gravity and manipulator are static; force analysis is shown in Figure1. In order to apperceive whole system, force/torque sensor is assembled on the base and close to the first joint. The force/torque sensor could measure force in three directions and torque rotated of three coordinate axes.Figure 1
Collision analysis.In Figure1, point C means collision point, which is assumed on ith joint. The center of force/torque sensor is expressed as B, which is set as origin of B coordinate system. Every joint of manipulator is built coordinate system as Ti according to DH parameter method. Ti represents coordinate system of ith joint, and p, f, and τ, respectively, mean collision force, measured force, and measured torque. nc represents normal vector of point c. Tc could be achieved by translating coordinate system Ti and no rotation. Then coordinate of point C in TB is obtained by coordinate transformation, as shown in the following:(1)TBc=TB1T12⋯Ti-1iTic,where(2)Tii-1=cθi-sθicαisθisαiaicθisθicθicαi-cθisαiaisθi0sαicαidi0001,Tic=100xic010yic001zic0001.When collision occurs, the outside force isp=(fcx,fcy,fcz)T, and data of force/torque sensor are f=(fx,fy,fz)T and τ=(τx,τy,τz)T. According to force analysis, data of outside force in TB should be equal to measured data in ideal condition, as shown in the following:(3)f=fcB,τ=τcB,where(4)fcB=fcBx,fcBy,fcBzT=Tp-TTBc3,1T,(5)τcB=τcBx,τcBy,τcBzT=cB×p=Tc-iTBc3,1×p.In (4),(6)p-=fcx,fcy,fcz,1Tand T(·)3,1 means extraction of elements of the first three rows and the first column.In (5),(7)c-i=xci,yci,zci,1T.Considering factor of gravity and manipulator dynamics in real world, general equation could be shown as follows:(8)f=fcB+fGB+fdBτ=τcB+τGB+τdB.In (8), fcB and τcB, respectively, express force and torque caused by collision mapping in basic coordinate system B; fGB and τGB, respectively, mean force and torque caused by gravity mapping in basic coordinate system B, which will be analyzed in Section 3; fdB and τdB, respectively, express force and torque caused by manipulator dynamics mapping in basic coordinate system B, which will be shown in Section 4.In (8), fGB, τGB, fdB, and τdB could be calculated based on states of manipulator and joint physical properties, such as joint angle, joint velocity, joint acceleration, and joint quality. Collision information is just contented in fcB and τcB, that is, the parameters of fcx, fcy, fcz, xci, yci, and zci.


## 2.2. Physical Constraint

Normal geometrical surface contains ellipsoid, sphere, cylindrical surface, and plane. They are able to construct all simple surfaces including outline of manipulators. Mathematical description mode of all manipulator surface is achieved through coordinate transformation. From the role of geometrical change, the core of four types of surface is ellipsoid, which is able to change to other three types by changing parameters, as shown in Figure2.Figure 2
Unified modeling method for basic geometric surface.Assume three axis lengths of ellipsoid are2α, 2β, and 2γ, setting center as origin of coordinate system. When α=β=γ, sphere will be formed; when γ→+∞, cylindrical surface is achieved; and when α→+∞, β→0, and γ→∞, plane surface is displayed. In summary, it just needs to build ellipsoid mode to describe all surface, as shown in the following:(9)Sc=cTATAc=1,where(10)A=1α0001β0001γ.Normal vector of pointC is set as nc and could be calculated with equation as follows:(11)nc=∇Sc∇Sc=2A2c2A2c=A2cA2c,where c=(x,y,z) means coordinate values and ∇ represents gradient.


## 3. Gravity Compensation

In the above section, gravity element is sampled asfGB and τGB. In this section, gravity influence on sensor will be analyzed, which relates to joint length, joint angle gravity center of joint, and quality. All joints cause force and torque on sensor, so each of them needs to be calculated. Force analysis is shown in Figure 3 in gravity.Figure 3
Gravity analysis.In Figure3, Di=xdi,ydi,zdiT is gravity center position in Ti coordinate; then ri means coordinate of gravity center of ith joint and is expressed as(12)ri=TTBiD-i3,1T,where(13)D-i=xdi,ydi,zdi,1T.Assuming variety of sensors isfGB∈R3, τGB∈R3, then (14) and (15) are achieved:(14)fGB=G0+G1+G2+⋯+Gi+⋯+Gn=∑i=0nGi,(15)τGB=r0×G0+r1×G1+⋯+ri×Gi+⋯+rn×Gn=G^0r0+G^1r1+⋯+G^iri+⋯+G^nrn=∑i=0nG^iri.In (14), Gi=(0,0,Gi)T (i≤n) and n is the total number of joints.In (15), in order to easily calculate with matrix, vector Gi is transformed into matrix G^i:(16)G^i=00-Gi000Gi00,i≤n.


## 4. Dynamic Compensation

In Section3, manipulator dynamic is sampled as fdB and τdB. In this section, dynamic factor is analyzed. When manipulator works, it also causes reaction force to sensor, so calculation is also needed. Dynamic force is very complex, which relates to joint length, joint angle, joint angular velocity, joint angular acceleration, and so on. In Figure 4, dynamic force is analyzed.Figure 4
Dynamic force analysis.In Figure4, assume αi is joint angular acceleration and Di=xdi,ydi,zdiT is gravity center position in Ti coordinate. Then, acceleration of gravity center is expressed as(17)ai=αi×Di.InTi coordinate, in order to keep moving as in Figure 4, the force and torque are calculated as shown in the following: (18)fdi=miai=miai×Di=mia^iDi,ταdi=Iiαi.In (18), mi and Ii, respectively, express quality and inertia of ith joint, and (19)a^i=0-aizaiyaiz0-aix-aiyaix0.Due to the fact thatTi is a motional coordinate, Coriolis force emerges in the whole system, which is expressed as(20)fkdi=2mivTi×ωTi.In (20), ωTi and vTi, respectively, mean velocity vector and angular velocity vector of Ti coordinate relative to basic coordinate.Based on the principle of equivalence, the above forces and torque ofi joint are transfered to B, which could be expressed as(21)ffBdi=TfdiTTBD3,1T,fkBdi=TfkdiTTBi3,1T,τfBdi=ffBdi×DBi=f^fBdiTTBiD-i3,1T,τkBdi=fkBdi×Ti=f^kBdiTTBi3,1T,ταBdi=ταdi.In the above equation,ffBdi and fkBdi, respectively, mean acceleration force and Coriolis force in the basic coordinate; τfBdi and τkBdi express torque of force transformation; torque could transfer with no other changes, so ταBdi=ταdi.Manipulator contains many joints, so all effect of force of joints on the basic coordinate is shown as(22)fdB=∑i=1nfBdi=∑i=1nffBdi+fkBdi,τdB=∑i=1nτBdi=∑i=1nτfBdi+τkBdi+ταBdi.


## 5. Solvability Analysis

At present, problem will be analyzed under the following limits in this subsection:(a)
Material of manipulator is rigid, which means no deformation produce.(b)
Collision is caused by manipulator and point of object.(c)
All sensor is ideal.(d)
Gravity is ignored in this subsection, because it has to be calculated in Section5.3.(e)
Manipulator dynamic is discussed in Section5.4, so assume manipulator is static and motion of outside object causes collision.In order to ensure solvability and uniqueness, four constraints are provided and compatibility principle is proposed.


### 5.1. Force Constraint

From Section2, combine (8) and (9); seven equations are achieved with six variables. Moreover, fGB, τGB, fdB, and τdB are zero under the above assumption.In order to easily calculate with matrix, vectorp is transformed into matrix p^:(23)p^=0-fczfcyfcz0-fcx-fcyfcx0.Then, part of (8) could be changed to(24)τcB=p^Tc-iTBc3,1.Combining (1), (8), and (24), the following equation is achieved:(25)fτ=MTp-TTBc3,1TTc-iTBc3,1,where(26)M=1000000100000010000000-fczfcy000fcz0-fcx000-fcyfcx0.M is relation matrix and it is singular matrix in ideal condition, that is, rank(M) = 5. A series of solutions about (24) are obtained. There are many states that make rank(M) = 6, such as noise of sensor, precision of sensor, and deformation. This problem will be solved in Section 5.5.


### 5.2. Geometric Constraint

Force act on the surface of manipulator, so mathematical direction of whole outline is necessary to constraint solution, that is,(27)Sc=0.S(·) means mathematical direction of manipulator surface.


### 5.3. Normal Vector Constraint

Because that force can only apply to manipulator from outside, the angle of force direction and normal vector is less thanπ/2. Based on this constraint, equation could be shown as(28)pTn<0.


### 5.4. Current Mutation Constraint

Strange geometries of manipulator surface and multiple degrees of freedom of manipulator may cause different force action with the same data of force/torque sensor. In this condition, uniqueness solution could not be calculated, so current mutation constraint is drawn in.Assuming collision point is on theith joint, then the current of this joint will change suddenly. In addition, the current of kth (k-i<0) joint also produces mutation, but the current of kth (k-i>0) joint will not change suddenly. In this law, collision point will be judged on the ith joint.On the one hand, this constraint avoids multiple solutions, and on the other hand, it could ensure the range of collision point and accelerate the speed of calculation.


### 5.5. Compatibility Solution Analysis

In Section5.1, if rank(M) = 5, then uniqueness solution will be achieved combining with (9) which means compatibility is the best. But many elements cause rank(M) = 6, such as noise of sensor, precision of sensor, and deformation. There is no solution through combining (8) and (9), so compatibility solution analysis is proposed to ensure an approximate solution. In Figure 5, process of method is shown.Figure 5
Process of compatibility.From Figure5, when rank(M) = 6, five equations will be chosen from matrix equation (25) and combining with (9); then one group of solution is obtained. Other five groups of solutions will be also achieved by the same way. Every solution includes the information about position and force size in three coordinate axes. Six normal vectors about solution position are achieved through (25). Then singular solutions should be removed by deviation of positions and normal vectors. At this moment, average values of the rest of solutions are calculated, and choose the closest point in outline of manipulator as approximate solution. At last, force size in three coordinate axes will be achieved by (25). This method avoids no solution condition and achieves an approximate solution.


## 5.1. Force Constraint

From Section2, combine (8) and (9); seven equations are achieved with six variables. Moreover, fGB, τGB, fdB, and τdB are zero under the above assumption.In order to easily calculate with matrix, vectorp is transformed into matrix p^:(23)p^=0-fczfcyfcz0-fcx-fcyfcx0.Then, part of (8) could be changed to(24)τcB=p^Tc-iTBc3,1.Combining (1), (8), and (24), the following equation is achieved:(25)fτ=MTp-TTBc3,1TTc-iTBc3,1,where(26)M=1000000100000010000000-fczfcy000fcz0-fcx000-fcyfcx0.M is relation matrix and it is singular matrix in ideal condition, that is, rank(M) = 5. A series of solutions about (24) are obtained. There are many states that make rank(M) = 6, such as noise of sensor, precision of sensor, and deformation. This problem will be solved in Section 5.5.


## 5.2. Geometric Constraint

Force act on the surface of manipulator, so mathematical direction of whole outline is necessary to constraint solution, that is,(27)Sc=0.S(·) means mathematical direction of manipulator surface.


## 5.3. Normal Vector Constraint

Because that force can only apply to manipulator from outside, the angle of force direction and normal vector is less thanπ/2. Based on this constraint, equation could be shown as(28)pTn<0.


## 5.4. Current Mutation Constraint

Strange geometries of manipulator surface and multiple degrees of freedom of manipulator may cause different force action with the same data of force/torque sensor. In this condition, uniqueness solution could not be calculated, so current mutation constraint is drawn in.Assuming collision point is on theith joint, then the current of this joint will change suddenly. In addition, the current of kth (k-i<0) joint also produces mutation, but the current of kth (k-i>0) joint will not change suddenly. In this law, collision point will be judged on the ith joint.On the one hand, this constraint avoids multiple solutions, and on the other hand, it could ensure the range of collision point and accelerate the speed of calculation.


## 5.5. Compatibility Solution Analysis

In Section5.1, if rank(M) = 5, then uniqueness solution will be achieved combining with (9) which means compatibility is the best. But many elements cause rank(M) = 6, such as noise of sensor, precision of sensor, and deformation. There is no solution through combining (8) and (9), so compatibility solution analysis is proposed to ensure an approximate solution. In Figure 5, process of method is shown.Figure 5
Process of compatibility.From Figure5, when rank(M) = 6, five equations will be chosen from matrix equation (25) and combining with (9); then one group of solution is obtained. Other five groups of solutions will be also achieved by the same way. Every solution includes the information about position and force size in three coordinate axes. Six normal vectors about solution position are achieved through (25). Then singular solutions should be removed by deviation of positions and normal vectors. At this moment, average values of the rest of solutions are calculated, and choose the closest point in outline of manipulator as approximate solution. At last, force size in three coordinate axes will be achieved by (25). This method avoids no solution condition and achieves an approximate solution.


## 6. Simulation Experiment

One manipulator with three degrees of freedom is established, as shown in Figure6. In order to simplify analysis, the outline of manipulator is composed of cylindrical surface and semisphere, and the size is set to rR. In addition, the material of every joint is uniform and set as 1060 alloys (ρ=2700 kg/m3) and geometry of every joint is central symmetry; then gravity center of joint lies on central axis.Figure 6
Manipulator structure.When collision occurs, data of force/torque sensor change with time. In engineering application, some signal processing methods will be adopted to achieve effective data. In this paper, our attention is focused on the method of collision sensing, so in order to simplify analysis, assume that manipulator is static. Parameters of links are listed in Table1.Table 1
Parameters of manipulator.
Name
Quality (Kg)
Position of gravity center in local coordinate (mm)
Link 0
1.629
(0, 0, 60)
Link 1
1.786
(0, 0, 69.995)
Link 2
2.565
(86.256, 0, 0)
Link 3
3.180
(106.906, 0, 0)In Figure6, l1=120 mm, l2=150 mm, l3=200 mm, l4=200 mm, and rR=40 mm. Due to the fact that joint connected surface is complex, assume collision point is only on link rod. Surface of each joint is described in coordinate of itself as follows:Link 1:(29)x12+y12=rR2,0<z1≤l2-rR.Link 2:(30)y22+z22=rR2,rR≤x2≤l3-rR.Link 3:(31)y32+z32=rR2,rR≤x3≤l4-rR.Top of Link 3:(32)x42+y42+z42=rR2,x4>0.According to coordinate transformation matrix as shown in(33)T01=cos⁡θ1-sin⁡θ100sin⁡θ1cos⁡θ100001l10001,T12=cos⁡θ20-sin⁡θ20sin⁡θ20cos⁡θ200-10l20001,T23=cos⁡θ3-sin⁡θ30l3cos⁡θ3sin⁡θ3cos⁡θ30l3sin⁡θ300100001,T34=100l4010000100001,equations of links’ outline in basic coordinate will be obtained as shown in the following:Link 1:(34)x1cos⁡θ1-y1sin⁡θ12+x1sin⁡θ1-y1cos⁡θ12=rR2,l1<z1≤l1+l2-rR.Link 2:(35)x2A+y2B2+l1+l22=rR2,l1C<x2C-y2B≤l1+l2-rRC.Link 3:(36)x3cθ3A+sθ3C-y3cθ3C-sθ3A+l3sθ3C+l3cθ3A2+l1+l2+z32=rR2,rRcθ3C-sθ3A-l3sθ3A+l3cθ3C≤x3cθ3C-sθ3A-y3cθ3A+sθ3C-l3sθ3A+l3cθ3C≤l4-rRcθ3C-sθ3A-l3sθ3A+l3cθ3C.Top of Link 3:(37)l4C-sθ3D+x4cθ3C-sθ3A-y4cθ3A+sθ3C-l3sθ3A+l3cθ3C2+l4cθ3A+sθ3C+x3cθ3A+sθ3C-y3cθ3C-sθ3D+l3sθ3C+l3cθ3D2+l1+l2+z42=rR2,x4cθ3C-sθ3D>0,where(38)A=cθ1sθ2+sθ1cθ2,B=cθ1cθ2+sθ2sθ1,C=cθ1cθ2-sθ2sθ1,D=cθ1sθ2-sθ1cθ2.Considering gravity compensation, the manipulator is analyzed as shown in Figure7, where a1=86.256 mm and a2=106.906 mm, (39)r1=a1cos⁡θ1cos⁡θ2,a1sin⁡θ1cos⁡θ2,l1+l2+a1sin⁡θ2T,r2=a1cos⁡θ1cos⁡θ2+a2cos⁡θ1cos⁡θ3,a1sin⁡θ1cos⁡θ2+a2sin⁡θ1cos⁡θ3,l1+l2+a1sin⁡θ2+a2sin⁡θ3T.Figure 7
Gravity compensation analysis.Mode of manipulator is built in ADAMS, and two groups of data are tested to verify our algorithm.Assumeθ1=π/6, θ2=-π/6, and θ3=-π/4, and 100 N force applies on the surface. Ten points are tested, throughout the entire surface.In another experiment, assumeθ1=π/6, θ2=-π/6, and θ3=-π/4, but force size is changed at the same position C = (206.08 72.79525.46).In Tables2 and 3, force size, force direction, and force position are known and set as input conditions; detected force and detected torque express the data of force/torque sensor, which could be achieved by measurement; calculated position and calculated force could be obtained by (8) and coordinate transformation with a series of calculations; force size error, force direction error, and position error are put forward to judge accuracy and validity of the calculation. Equation of each error is shown as follows:Force size error:(40)ΔS=fsx2+fsy2+fsz2-fcBx2+fcBy2+fcBz2fcBx2+fcBy2+fcBz2.Force direction error:(41)Δθ=arccos⁡fsxfcBx+fsyfcBy+fszfcBzfsx2+fsy2+fsz2fcBx2+fcBy2+fcBz2.Position error:(42)ΔP=xcB-xsB+ycB-ysB+zcB-zsBxcB2+ycB2+zcB2.Table 2
Different point experiment.
Number
Force size (N)
Force direction (N)
Force position (mm)
Detected force (N)
Detected torque (N·m)
Calculated position (mm)
Calculated force (mm)
Force size  error (%)
Force direction error (°)
Position error (%)
1
100
93.81529.9627 33.158
400 150
−93.815279.8145 −33.158
−25.4134−12.746 6.654
40.340.05 152.38
93.147310.2345 32.153
0.930
2.087
1.78
2
100
60.913874.9999 25.7783
040 175
−60.918314.7773 −25.7783
−14.8139−10.6599 9.4891
2.1239.83 176.33
59.233274.5389 25.2234
1.507
2.657
2.017
3
100
−90.441742.0904 −6.9776
400 200
90.441747.6868 6.9776
−18.489717.8092 5.3689
42.981.46 197.94
−89.134740.6234 −7.7463
1.739
3.069
3.187
4
100
73.595810.5043 −66.8828
65.0237.54 267.16
−73.595879.2729 66.8828
−21.5907−24.0106 9.1323
68.5840.24 263.34
71.37489.0657 −66.1635
2.255
4.019
3.632
5
100
−4.786994.594 −32.0789
83.752.17 312.5
4.78694.8169 32.0789
2.7225−1.1907 −0.8801
85.632.37 324.49
−5.037596.4736 −34.785
2.677
4.867
4.349
6
100
−36.4847−68.9913 62.5226
118.6668.51 302.92
36.4847158.7685 −62.5226
−52.0918.4709 12.7395
124.9463.89 306.35
−35.8134−70.7645 65.386
2.789
5.032
4.310
7
100
−81.535344.681 36.8185
186.9361.74 442.98
81.535345.0962 −36.8185
−9.388243.001 −6.3337
174.6568.49 452.76
−83.475644.9652 39.8583
2.853
5.582
5.943
8
100
79.821538.4423 −46.3758
210.36121.45 475.56
−79.821551.3349 46.3758
−2.9938−47.7155 8.6602
222.08131.75 487.74
82.946740.8643 −45.7643
3.171
6.111
6.404
9
100
−80.100658.8915 10.7551
206.0872.79 525.46
80.100630.8856 −10.7551
3.254544.3061 −10.9144
219.7984.86 539.75
−84.475458.4393 10.4637
3.251
7.457
7.041
10
100
−74.0921−18.1331 −64.665
222.17128.27 587.74
74.0921107.9102 64.665
−29.270729.1803 1.5773
238.57113.45 572.84
−76.4836−21.9574 −68.4833
4.985
7.886
7.192Table 3
Different force experiment.
Number
Force size (N)
Force direction (N)
Force position (mm)
Detected force (N)
Detected torque (N·m)
Calculated position (mm)
Calculated force (mm)
Force size error (%)
Force direction error (°)
Position  error (%)
1
60
−48.060435.3349 6.4531
206.0872.79 525.46
48.060454.4423 −6.4531
−8.810426.5837 −3.7276
223.5385.34 542.45
−46.468432.2344 7.1736
2.993
8.384
8.257
2
80
−64.080547.1132 8.6041
206.0872.79 525.46
64.080542.664 −8.0641
−2.77793504449 −7.321
223.4480.32 543.24
−60.367446.6237 9.7344
3.106
7.721
7.498
3
100
−80.100658.8915 10.7551
206.0872.79 525.46
80.100630.8856 −10.7551
3.254544.3061 −10.9144
219.7984.86 539.75
−84.475458.4393 10.4637
3.251
7.457
7.041
4
120
−96.120870.6698 12.9062
206.0872.79 525.46
96.102819.1073 −12.9062
9.28753.1673 −14.5078
216.4587.87 535.64
−99.834870.7982 10.5374
2.843
7.060
6.261
5
140
−112.140982.4482 15.0572
206.0872.79 525.46
112.14097.329 −15.0572
15.319462.0285 −18.1011
213.2385.56 538.97
−113.453287.1726 13.2736
3.690
6.620
5.874
6
160
−128.16194.2265 17.2082
206.0872.79 525.46
128.161−4.4493 −17.2082
21.351970.8898 −21.6945
213.7481.46 539.53
−123.756295.6352 17.3845
2.634
6.104
5.342
7
180
−144.1812106.0048 19.3593
206.0872.79 525.46
144.1812−16.2276 −19.3593
27.384379.751 −25.2879
217.3676.87 540.37
−149.2352104.8453 18.4632
3.315
5.726
5.319
8
200
−160.2013117.7831 21.5103
206.0872.79 525.46
160.2013−28.0059 −21.5103
33.416888.6122 −28.8813
223.4278.36 530.75
−155.7297119.4328 20.5643
2.671
5.201
4.955
9
220
−176.2214129.5614 23.6613
206.0872.79 525.46
176.2214−39.7842 −23.6613
39.449397.4734 −32.4747
197.4565.32 535.75
−171.2194131.3342 22.9472
2.995
5.172
4.637
10
240
−192.2415141.3397 25.8123
206.0872.79 525.46
192.2415−51.5625 −25.8123
45.4817106.3346 −36.068
199.6365.37 533.74
−196.24387140.3484 29.6547
3.082
4.973
3.892In these three equations,fsx, fsy, and fsz mean force size of calculated force and xsB, ysB, and zsB mean calculated position mapping to three axes in basic coordination; fcBx, fcBy, and fcBz mean force size of force and xcB, ycB, and zcB mean calculated position mapping to three axes in basic coordination.From Tables2 and 3, collision detection is effective and the relative error is extremely small in simulation environment. In order to intuitively express data relation, Figures 8 and 9 are drawn to show them. In Figure 8, relation between different collision positions and each error is established. In Figure 9, relation between different force sizes and each error is established.Figure 8
Relation between collision positions and errors.Figure 9
Relation between force and errors.In Figure8, force size error, force direction error, and position error all become larger with farther distance. Different collision point produces different error and each error will be larger with farther distance. If we use this method to test collision information, it must ensure whether the biggest errors meet accuracy requirements at the farthest point of manipulator.In Figure9, force direction error and position error become smaller with bigger force, and force size error is fluctuating. Sensitivity of this method is almost certain at the same point, so force direction error and position error become smaller with bigger force. About fluctuating of force size error, (40) has given our explanation that numerator and denominator influence force size error at the same time and have the same trend.


## 7. Summary

Both animal’s arm and person’s arm are wrapped by skins, which help them protect their biological organization and apperceive environmental information, such as temperature change and object collision. Researchers except manipulators also have the same function with biological arm, but the method is expensive and easily damaged by distributing sensors on the whole manipulator’s outline, so one indirect method to detect collision point and direction by using force/torque sensor is put forward. Compared to skin force sensor, the force/torque sensor is easy to be installed and maintained. In our method, constraint equations and compatibility solution analysis is used to ensure existence of solution. Moreover, gravity compensation and dynamic compensation are also described, which indicate this theory could be effective in gravity environment and when manipulators are exercising. At last, the simulation experiment is executed to verify our algorithm and the result is perfect.But there exist a lot of engineering problems, if this method is used in real system, which could be as follows:(1) how to extract more accurate force and torque information, based on the fact that signal of sensor is fluctuant; (2) how to describe manipulator’s outline accurately, because it has complex shape; (3) how to measure force and torque in an instant when manipulator collides; (4) how to realize real-time perception, on the condition that this theory contents a lot of calculations. In order to promote the theory in engineering application some experiments will be done on a real system, especially putting focus on the above problems.



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*Source: 6291216-2015-12-14.xml*