CN111625012A - Distributed cooperative operation method for multi-space robot - Google Patents

Abstract

The invention relates to a distributed cooperative operation method for multiple space robots, which realizes stable control of a combined system by using information interaction among robots. The method is suitable for different network-like connection topologies among the robots, avoids the requirement of a central node, and can flexibly increase or reduce the number of the robots in the system and the connection configurations thereof. Compared with the conventional spacecraft control method, the method has the following advantages: 1) the method is a distributed algorithm, and the robot unit control moment calculation, control parameter coordination and updating do not need a central unit, so that the flexibility and robustness of the system are improved; 2) the method is suitable for various connection topologies such as a network topology structure, a bus topology and the like, and has wide practicability; 3) the method is suitable for distributed cooperative control among heterogeneous robots, and is suitable for thrustor robots and reaction flywheel robots.

Description

A distributed cooperative operation method for multi-space robots

technical field

The invention belongs to the field of spacecraft control, and relates to a distributed cooperative operation method of multi-space robots, in particular to a method for realizing stable control of a combination of multiple space robots through distributed computing.

Background technique

With the development of aerospace technology, spacecraft play an increasingly important role in communication, navigation and positioning, and earth observation. The direct and indirect losses caused by the failure of spacecraft on-orbit are huge. The development of on-orbit service technology is the key development direction of the world's aerospace powers. To this end, different researchers have developed various types of space robots, such as space robots based on space manipulators, space tethered robots based on tethers, space cell robots based on cellular concepts, and space flying net robots based on rope nets. . For a large failed spacecraft, its mass is as high as several tons. After multiple small space robots are docked with the failed spacecraft, information interaction and coordination are required to implement takeover control of the failed spacecraft to achieve stable control of the entire combined system. Different from the single robot system, the structure of this system is distributed, so it is necessary to design a distributed cooperative control scheme suitable for its system structure.

In order to solve the collaborative control of the combined system by multiple space robots, the invention provides a distributed collaborative control method, which utilizes the information interaction between the robots to realize the stable control of the combined system. This method is suitable for different network-like connection topologies between robots, avoids the need for a central node, and can flexibly increase or decrease the number of robots and their connection configurations in the system.

SUMMARY OF THE INVENTION

technical problem to be solved

In order to avoid the deficiencies of the prior art, the present invention proposes a distributed collaborative operation method for multi-space robots, which realizes distributed collaborative control among multiple robots, that is, utilizes multiple robots with control capability to stabilize the combined system. control.

Technical solutions

其中各元素分别表示机器人i的接口1至接口n_i相连的机器人的ID值，起始值为0；操作步骤如下：A multi-space robot distributed collaborative operation method, characterized in that: there are N robots in the system, each robot has a unique identification ID, the set of IDs of all robots is X, and ID=0 is defined as an illegal identification; The robot whose identity is ID _i is robot i; robot i records the ID list of all connected robots in the system, denoted as

Each element represents the ID value of the robot connected to the interface 1 of the robot i to the interface n _i , and the initial value is 0; the operation steps are as follows:

Step 1. Connection topology detection and update: each robot j∈X performs connection topology detection and update every T seconds, where T≥DΔt, D is the diameter of the multi-robot link topology undirected graph, which is determined by the robot network topology; Δt is the information interaction time interval between robots, each robot actively sends connection detection information or responds to the connection detection information of adjacent robots;

The process of robot j actively sending connection detection information:

Robot j adopts the general knowledge method in the field of network communication, and the method sends connection requests through all data interfaces of itself, and the transmitted information includes the identification ID _j of this robot; if the interface p of robot _j , 1≤p≤nj receives adjacent In the connection detection information of the robot's reply, the identifier of the robot connected to the interface p is read as ID _k , then l _jp =ID _k ; if the interface p does not receive a reply, it means that the interface p is not connected to the robot Or the connection robot fails, then let l _jp = 0;

The process of replying to the connection detection information of adjacent robots:

The interface r of the robot j receives the connection detection information of the adjacent robot, reads the ID _k of the robot connected to the interface r contained therein, and sets L _jr =ID _k , and returns its own connection detection information through the interface r ; The interface r of robot j receives the connection detection information of the adjacent robot, reads the identification of the robot connected to the interface r contained in it as ID _k , makes L _jr =ID _k , and returns its own connection detection through the interface r information;

This step is continuously executed in a timed cycle;

Step 2. Initialize control parameters:

The set of all robots that receive ground remote control commands is C _com , and the set size is N _com 1≤N _com ≤N;

All robot IDs _l ∈ C _com receive the desired attitude σ _d sent by the ground measurement and control station, and record the receiving time T _l ;

Robot l connects the list L _l according to its own information, and sends σ _d and T _l to all neighboring robots m in L _l . After robot m receives σ _d and T _l , it compares T _l and T _m ;

If T _l > T _m , update the σ _d recorded by itself, let T _m =T _l , and forward the σ _d and the update time T _m to all neighboring robots except robot l; otherwise ignore;

Step 3. Calculate the control torque of all robots:

Robot ID _ξ ∈ X uses sensors to measure spacecraft attitude σ _ξ ∈ R ³ , angular velocity ω _ξ ∈ R ³ and angular acceleration expressed by modified Rodrigues parameters

There are three groups of control parameters for each robot:

参数D_ξ＝diag(δ_ξ1,δ_ξ2,δ_ξ3)、K_ξ＝diag(κ_ξ1,κ_ξ2,κ_ξ3)，其中

δ_ξ1,δ_ξ2,δ_ξ3≥0，κ_ξ1,κ_ξ2,κ_ξ3＞0，所有机器人的控制参数初始值均设定为一致，并在步骤5进行更新；根据控制参数J_ξ、D_ξ、K_ξ和期望姿态σ_d计算机器人ξ的控制力矩τ_ξ，计算方式如下：include parameters

Parameters D _ξ =diag(δ _ξ1 ,δ _ξ2 ,δ _ξ3 ), K _ξ =diag(κ _ξ1 ,κ _ξ2 ,κ _ξ3 ), where

δ _ξ1 , δ _ξ2 , δ _ξ3 ≥0, κ _ξ1 , κ _ξ2 , κ _ξ3 > 0, the initial values of all robots’ control parameters are set to be the same, and are updated in step 5; according to the control parameters J _ξ , D _ξ , K _ξ and the desired attitude σ _d to calculate the control torque τ _ξ of the robot ξ, the calculation method is as follows:

Among them: σ _e is the attitude error under the modified Rodris parameter;

Step 4. Parameter scale factor calculation:

Each robot calculates its own parameter scale factor according to its own residual energy. For robot ID _ζ ∈X, the parameter scale factor W _ζ = [W _ζ1 W _ζ2 W _ζ3 ], where the value range of each element is [0 1], the calculation method as follows:

Case 1: If the robot ζ actuator is a thruster, its remaining propellant is calculated as:

Among them: A is a constant, the value range is 3~100; B is a constant, the value range is 1~10; e _ζ is the remaining propellant mass of the robot, and e _full is the total amount of the robot fuel storage tank;

Case 2: If the robot ζ actuator is three orthogonally mounted reaction flywheels, its residual energy is calculated as:

Among them: r _ζ1 , r _ζ2 , r _ζ3 are the rotational speeds of the three reaction flywheels of the robot ζ, respectively; r _full is the saturated rotational speed of the flywheel; α _ζ1 , α _ζ2 , α _ζ3 are calculated using the following formulas respectively:

T_ζ为机器人ζ飞轮配置矩阵，表示飞轮旋转轴坐标系与惯性坐标系的旋转矩阵；in:

T _ζ is the robot ζ flywheel configuration matrix, representing the rotation matrix of the flywheel rotation axis coordinate system and the inertial coordinate system;

Step 5. Control parameter update:

All robots regularly update control parameters and parameter scaling factors with neighboring robots. For robots

主动发送信息的具体方式如下：Case 1: Robot

The specific ways of actively sending information are as follows:

按照

记录的相邻机器人ID，依次与其相连的机器人进行数据交换：robot

according to

The recorded IDs of adjacent robots, in turn, exchange data with the robots connected to them:

Step (1): another q=1;

的值，若

则跳至步骤(3)；若

假设

表示机器人

的接口q和机器人π相连，机器人

将

发送给机器人π，机器人π回复自身的W_π，

δ_π1,δ_π2,δ_π3，κ_π1,κ_π2,κ_π3数据；完成信息交互后机器人

和机器人π分别按照下式更新自身的参数比例因子和控制参数Step (2): Judgment

value, if

Then skip to step (3); if

Assumption

Represents a robot

The interface q is connected to the robot π, the robot

Will

Send to robot π, robot π replies with its own W _π ,

δ _π1 ,δ _π2 ,δ _π3 ,κ _π1 ,κ _π2 ,κ _π3 data; after completing the information interaction, the robot

and robot π to update their own parameter scale factors and control parameters according to the following formulas respectively

则返回步骤3，否则进行步骤(2)；Step (3): q=q+1, if

Then return to step 3, otherwise go to step (2);

接收到相邻机器人的信息，处理方式和步骤(2)中的机器人π的处理步骤一致；Case 2: Robot

After receiving the information of the adjacent robot, the processing method is consistent with the processing steps of the robot π in step (2);

Return to step 3 after this step is completed.

其中：B为常数，取值范围为1～10；e_ζ为机器人剩余推进剂质量，e_full为机器人燃料储箱的总量。Case 1 in Step 4: If the robot ζ actuator is a thruster, the remaining propellant is calculated as:

Among them: B is a constant, the value range is 1~10; e _ζ is the remaining mass of the propellant of the robot, and e _full is the total amount of the robot's fuel storage tank.

Case 2 in Step 4: If the robot ζ actuator is three orthogonally installed reaction flywheels, its residual energy is calculated as:

其中：

T_ζ为机器人ζ飞轮配置矩阵，表示飞轮旋转轴坐标系与惯性坐标系的旋转矩阵。Among them: r _ζ1 , r _ζ2 , r _ζ3 are the rotational speeds of the three reaction flywheels of the robot ζ, respectively; r _full is the saturated rotational speed of the flywheel; α _ζ1 , α _ζ2 , α _ζ3 are calculated using the following formulas respectively:

in:

T _ζ is the configuration matrix of the robot ζ flywheel, which represents the rotation matrix of the flywheel rotation axis coordinate system and the inertial coordinate system.

beneficial effect

The distributed cooperative operation method of multi-space robots proposed by the invention utilizes the information interaction between robots to realize stable control of the combined system. This method is suitable for different network-like connection topologies between robots, avoids the need for a central node, and can flexibly increase or decrease the number of robots and their connection configurations in the system. Compared with the conventional spacecraft control method, it has advantages in the following aspects: 1) This method is a distributed algorithm, and the robot unit control torque calculation, control parameter coordination and update do not require a central unit, which improves the flexibility and robustness of the system 2) This method is suitable for various connection topologies such as network topology and bus topology, and has wide practicability; 3) This method is suitable for distributed collaborative control between heterogeneous robots, and can be applied to thruster robots and reaction flywheel robots .

Description of drawings

Figure 1: Topology Probing and Updating Data Connection Relationships

Figure 2: Control parameter initialization data connection relationship

Figure 3: Robot information interaction and parameter update

Detailed ways

The present invention will now be further described in conjunction with the embodiments and accompanying drawings:

其中各元素分别表示机器人i的接口1至接口n_i相连的机器人的ID值，起始值为0。Suppose there are N robots in the system, each robot has a unique ID, the set of IDs of all robots is X, and ID=0 is defined as an illegal ID; let the robot with ID _i be robot i, and there are at most n _i interfaces, n _i ≥ 1, the specific number is determined by the robot structure design. Robot i records the ID list of all connected robots in the system, denoted as

Each element respectively represents the ID value of the robot connected to the interface 1 of the robot i to the interface n _i , and the initial value is 0.

To achieve the above object, the technical solution adopted in the present invention comprises the following steps:

Step 1: Connect topology detection and update

The purpose of this step is to establish a connection with the adjacent robot by detecting the adjacent robot, and update the connection relationship regularly. This step is continuously executed in a timed loop.

Each robot j∈X performs connection topology detection and update every T seconds, where T≥DΔt, D is the diameter of the multi-robot link topology undirected graph, which is determined by the robot network topology; Δt is the information interaction time interval between robots , each robot needs to actively send connection detection information or reply to the connection detection information of adjacent robots.

Among them, the process of robot j actively sending connection detection information is (1), and the process of replying to the connection detection information of adjacent robots is (2):

(1) Robot j actively sends connection detection information to neighboring robots: it sends connection requests through all its own data interfaces, and the transmitted information at least contains the identification ID _j of the robot. Repeat. If the interface p (1≤p≤n _j ) of the robot j receives the connection detection information replied by the adjacent robot, let the ID k of the robot connected to the interface p read therein be ID _k , then let l _jp = ID _k ; if the interface p (1≤p≤n _j ) does not receive a reply, it means that the interface p is not connected to the robot or the connected robot fails, then let l _jp =0.

(2) The robot j replies to the connection detection information of the adjacent robot: if the interface r of the robot j receives the connection detection information of the adjacent robot, let the ID of the robot connected to the interface r contained in it read be ID _k , let L _jr =ID _k , and reply its own connection detection information through interface r.

Step 2: Initialize control parameters

The purpose of this step is to receive ground remote control commands, and gradually update the control expectations of all robots through information interaction between robots.

Let the set of all robots that can receive ground remote control commands be C _com , and the set size is N _com (1≤N _com ≤N).

For all robot IDs _l ∈ C _com , they receive the desired attitude σ _d sent by the ground measurement and control station, and record the receiving time T _l .

Robot l connects the list L _l according to its own information, and sends σ _d and T _l to all neighboring robots in L _l . Might as well set a certain adjacent robot as m, after robot m receives σ _d and T _l , compares T _l and T _m .

If T _l >T _m , update the self-recorded σ _d , let T _m =T _l , and forward σ _d and the update time T _m to all neighboring robots except robot 1; otherwise, ignore.

Step 3: Calculate the control torque

根据自身测量数据和控制参数计算控制力矩：All robots perform control torque calculation. For robot ID _ξ ∈X, let it use sensors to measure spacecraft attitude σ _ξ ∈ R ³ , angular velocity ω _ξ ∈ R ³ and angular acceleration expressed by modified Rodrigues parameters

Calculate the control torque according to its own measurement data and control parameters:

There are three groups of control parameters for each robot:

参数D_ξ＝diag(δ_ξ1,δ_ξ2,δ_ξ3)、K_ξ＝diag(κ_ξ1,κ_ξ2,κ_ξ3)，其中

δ_ξ1,δ_ξ2,δ_ξ3≥0，κ_ξ1,κ_ξ2,κ_ξ3＞0，所有机器人的控制参数初始值均设定为一致，并在步骤五进行更新。根据控制参数J_ξ、D_ξ、K_ξ和期望姿态σ_d计算机器人ξ的控制力矩τ_ξ，计算方式如下：include parameters

Parameters D _ξ =diag(δ _ξ1 ,δ _ξ2 ,δ _ξ3 ), K _ξ =diag(κ _ξ1 ,κ _ξ2 ,κ _ξ3 ), where

δ _ξ1 , δ _ξ2 , δ _ξ3 ≥0, κ _ξ1 , κ _ξ2 , κ _ξ3 >0, the initial values of all the robot control parameters are set to be the same, and are updated in step 5. Calculate the control torque τ _ξ of the robot ξ according to the control parameters J _ξ , D _ξ , K _ξ and the desired attitude σ _d , and the calculation method is as follows:

Among them, σ _e is the attitude error expressed by the modified Rodris parameter, which is a general knowledge method in the industry and does not belong to the content of the present invention. The specific expression is:

Step 4: Parameter scale factor calculation

Each robot calculates its own parameter scale factor according to its own residual energy. For robot ID _ζ ∈X, the parameter scale factor W _ζ = [W _ζ1 W _ζ2 W _ζ3 ], where the value range of each element is [0 1]. The specific calculation method is as follows:

(1) If the robot ζ actuator is a thruster, one of the following two formulas (3) and (4) can be used to calculate the remaining propellant.

Among them, A is a constant with a value range of 3 to 100; B is a constant with a value range of 1 to 10; e _ζ is the remaining mass of the propellant of the robot, and e _full is the total amount of the robot's fuel tank.

(2) If the robot ζ actuator is three orthogonally installed reaction flywheels, its residual energy can be calculated by one of the following formulas (5) and (6)

Among them, r _ζ1 , r _ζ2 , r _ζ3 are the rotational speeds of the three reaction flywheels of the robot ζ, respectively; r _full is the saturated rotational speed of the flywheel; α _ζ1 , α _ζ2 , α _ζ3 are calculated by the following formulas respectively

T_ζ为机器人ζ飞轮配置矩阵，表示飞轮旋转轴坐标系与惯性坐标系的旋转矩阵，为业内通识方法，不再赘述。in

T _ζ is the robot ζ flywheel configuration matrix, which represents the rotation matrix of the flywheel rotation axis coordinate system and the inertial coordinate system, which is a common method in the industry, and will not be repeated here.

Step 5: Control parameter update

All robots regularly update control parameters and parameter scaling factors with neighboring robots. For robots

主动发送信息的具体方式如下：Case 1: Robot

The specific ways of actively sending information are as follows:

按照

记录的相邻机器人ID，依次与其相连的机器人进行数据交换：robot

according to

The recorded IDs of adjacent robots, in turn, exchange data with the robots connected to them:

Step (1): another q=1;

的值，若

则跳至步骤(3)；若

假设

表示机器人

的接口q和机器人π相连，机器人

将

发送给机器人π，机器人π回复自身的W_π，

δ_π1,δ_π2,δ_π3，κ_π1,κ_π2,κ_π3数据。完成信息交互后机器人

和机器人π分别按照下式更新自身的参数比例因子和控制参数Step (2): Judgment

value, if

Then skip to step (3); if

Assumption

Represents a robot

The interface q is connected to the robot π, the robot

Will

Send to robot π, robot π replies with its own W _π ,

δ _π1 , δ _π2 , δ _π3 , κ _π1 , κ _π2 , κ _π3 data. After completing the information interaction, the robot

and robot π to update their own parameter scale factors and control parameters according to the following formulas respectively

则返回步骤三，否则进行步骤(2)。Step (3): if q=q+1

Then go back to step 3, otherwise go to step (2).

接收到相邻机器人的信息，处理方式和步骤(2)中的机器人π的处理步骤一致。Case 2: Robot

When the information of the adjacent robot is received, the processing method is the same as that of the robot π in step (2).

After this step is completed, go back to step 3.

Claims (3)

1. A distributed cooperative operation method of a multi-space robot is characterized in that: the system comprises N robots, each robot has a unique identity ID, the set of the IDs of all the robots is X, and the defined ID is 0 and is an illegal identity; let the identity be ID_iThe robot of (1) is a robot i; the robot i records the ID list of all the connected robots in the system and records the ID list as

Wherein each element represents an interface 1 to an interface n of the robot i_iThe initial value of the ID value of the connected robot is 0; the operation steps are as follows:

step 1, connection topology detection and updating: each robot j belongs to X and performs connection topology detection and updating every T seconds, wherein T is more than or equal to D delta T, D is the diameter of an undirected graph of the multi-robot link topology and is determined by the robot network topology; delta t is the time interval of information interaction between the robots, and each robot actively sends connection detection information or replies the connection detection information of adjacent robots;

the process of the robot j actively sending the connection detection information:

the robot j adopts a network communication field identification method to send a connection request through all data interfaces of the robot, and the transmitted information comprises the identification ID of the robot_j(ii) a If the interface p of the robot j is more than or equal to 1 and less than or equal to n_jReceiving the connection detection information replied by the adjacent robot, reading the ID of the robot connected with the interface p contained in the connection detection information_kThen let l_jp＝ID_k(ii) a If the interface p does not receive the reply, it means that the interface p is not connected with the robot or the connected robot is invalid, and let l_jp＝0；

The process of replying the connection detection information of the adjacent robot comprises the following steps:

the interface r of the robot j receives the connection detection information of the adjacent robot, reads the ID of the robot connected with the interface r contained in the connection detection information_kLet L_jr＝ID_kAnd reply the connection detection information of oneself through the interface r; the interface r of the robot j receives the connection detection information of the adjacent robot, reads the ID of the robot connected with the interface r contained in the connection detection information_kLet L_jr＝ID_kAnd reply the connection detection information of oneself through the interface r;

the step is executed continuously in a timing cycle;

step 2, initializing control parameters:

the set of all robots for receiving ground remote control commands is C_comSet size N_com1≤N_com≤N；

All robot ID_l∈C_comReceiving expected attitude sigma sent by ground measurement and control station_dRecording the reception time T_l；

The robot L connects the list L according to the information thereof_lWill σ_dAnd T_lIs sent to L_lAll neighboring robots m in (1), robot m receives σ_dAnd T_lThen, T is compared_lAnd T_m；

If T_l＞T_mIf yes, then update the recorded sigma_dLet T_m＝T_lAnd will be_dAnd an update time T_mFurther forwarding to all adjacent robots except the robot l; otherwise, ignoring;

and 3, calculating the control moments of all the robots:

robot ID_ξ∈ X use sensors to measure spacecraft attitude σ in terms of modified rodgers parameters_ξ∈R³Angular velocity omega_ξ∈R³And angular acceleration

Each robot control parameter is three groups:

including parameters

Parameter D_ξ＝diag(_ξ1,_ξ2,_ξ3)、K_ξ＝diag(κ_ξ1,κ_ξ2,κ_ξ3) Wherein

_ξ1,_ξ2,_ξ3≥0，κ_ξ1,κ_ξ2,κ_ξ3If the initial values of the control parameters of all the robots are more than 0, the initial values of the control parameters of all the robots are set to be consistent, and the control parameters are updated in step 5; according to the control parameter J_ξ、D_ξ、K_ξAnd the desired attitude σ_dCalculating control moment τ of robot ξ_ξThe calculation method is as follows:

wherein: sigma_eCorrecting attitude errors under the representation of the Rodris parameter;

step 4, calculating parameter scale factors:

each robot calculates its own parameter scale factor according to its own residual energy, for robot ID_ζ∈ X, parameter scale factor W_ζ＝[W_ζ1W_ζ2W_ζ3]Wherein each element has a value range of [ 01 ]]The calculation method is as follows:

case 1: if the zeta executor of the robot is a thruster, the residual propellant is calculated as:

wherein: a is a constant and has a value range of 3-100; b is a constant, and the value range is 1-10; e.g. of the type_ζFor the remaining propellant mass of the robot, e_fullFor robot fuel storageThe total amount of bins;

case 2: if the zeta executor of the robot is three reaction flywheels which are installed in an orthogonal mode, the residual energy is calculated as follows:

wherein: r is_ζ1、r_ζ2、r_ζ3The rotating speeds of the three reaction flywheels of the robot are respectively zeta; r is_fullIs the saturation speed of flywheel α_ζ1，α_ζ2，α_ζ3Calculated using the following formula, respectively:

wherein:

T_ζconfiguring a matrix for a zeta flywheel of the robot, wherein the matrix represents a rotation matrix of a flywheel rotation axis coordinate system and an inertia coordinate system;

and 5, updating control parameters:

all robots regularly update control parameters and parameter scale factors with adjacent robots

Case 1: robot

The specific way of actively sending information is as follows:

robot

According to

Recorded neighboring robot IDs, in turn, therewithThe connected robots exchange data:

step (1): q is 1;

step (2): judgment of

A value of, if

Jumping to the step (3); if it is

Suppose that

Presentation robot

The interface q is connected with the robot pi, and the robot

Will be provided with

Sending the data to the robot pi, and the robot pi replies the W of the robot pi_π，

_π1,_π2,_π3，κ_π1,κ_π2,κ_π3Data; robot after finishing information interaction

And a machineThe human pi respectively updates the self parameter scale factor and the control parameter according to the following formula

And (3): q is q +1, if

Returning to the step 3, otherwise, performing the step (2);

case 2: robot

Receiving information of adjacent robots, wherein the processing mode is consistent with the processing step of the robot pi in the step (2);

and returning to the step 3 after the step is executed.

2. The distributed cooperative operation method of a multi-space robot according to claim 1, characterized in that: case 1 in step 4: if the zeta executor of the robot is a thruster, the residual propellant is calculated as:

wherein: b is a constant, and the value range is 1-10; e.g. of the type_ζFor the remaining propellant mass of the robot, e_fullThe total amount of the robot fuel storage tank.

3. The distributed cooperative operation method of a multi-space robot according to claim 1, characterized in that: case 2 in step 4: if the zeta executor of the robot is three reaction flywheels which are installed in an orthogonal mode, the residual energy is calculated as follows:

wherein: r is_ζ1、r_ζ2、r_ζ3The rotating speeds of the three reaction flywheels of the robot are respectively zeta; r is_fullIs the saturation speed of flywheel α_ζ1，α_ζ2，α_ζ3Calculated using the following formula, respectively:

wherein:

T_ζa matrix is configured for the zeta flywheel of the robot, and the matrix represents a rotation matrix of a flywheel rotation axis coordinate system and an inertia coordinate system.

Images