Fast estimation of space-robots inertia parameters: A modular mathematical formulation

Author’s Accepted Manuscript Fast estimation of space-robots inertia parameters;A Modular Mathematical Formulation Seyed Yaser Nabavi Chashmi, Seyed MohammadBagher Malaek www.elsevier.com/locate/actaastro

PII: DOI: Reference:

S0094-5765(15)30356-8 http://dx.doi.org/10.1016/j.actaastro.2016.04.037 AA5838

To appear in: Acta Astronautica Received date: 27 December 2015 Accepted date: 26 April 2016 Cite this article as: Seyed Yaser Nabavi Chashmi and Seyed Mohammad-Bagher Malaek, Fast estimation of space-robots inertia parameters;A Modular Mathematical Formulation, Acta Astronautica, http://dx.doi.org/10.1016/j.actaastro.2016.04.037 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Fast Estimation of Space-Robots Inertia Parameters; A Modular Mathematical Formulation Seyed Yaser Nabavi Chashmi, Seyed Mohammad-Bagher Malaek Department of Aerospace, Sharif University of Technology, Tehran, Iran [email protected] [email protected] Abstract This work aims to propose a new technique that considerably helps enhance time and precision needed to identify “Inertia Parameters (IPs)” of a typical Autonomous Space-Robot (ASR). Operations might include, capturing an unknown Target Space-Object (TSO), “active space-debris removal” or “automated in-orbit assemblies”. In these operations generating precise successive commands are essential to the success of the mission. We show how a generalized, repeatable estimation-process could play an effective role to manage the operation. With the help of the well-known Force-Based approach, a new “modular formulation” has been developed to simultaneously identify IPs of an ASR while it captures a TSO. The idea is to reorganize the equations with associated IPs with a “Modular Set” of matrices instead of a single matrix representing the overall system dynamics. The devised Modular Matrix Set will then facilitate the estimation process. It provides a conjugate linear model in mass and inertia terms. The new formulation is, therefore, well-suited for “simultaneous estimation processes” using recursive algorithms like RLS. Further enhancements would be needed for cases the effect of center of mass location becomes important. Extensive case studies reveal that estimation time is drastically reduced which in-turn paves the way to acquire better results.

Keywords Inertia Parameters, System Identification, Space-Robots, Space Debris Removal

1

Introduction

Newly proposed space missions are complex and involve precise in-orbit operations. Active Space-Debris Removal (ASDR) [1,2], cargo delivery using space elevators [3] autonomous space-station docking, in-orbit assembly, in-orbit maintenance, refueling and berthing are just few examples. In fact, ASRs are soon expected to respond to different operational demands [4]. In this work, however, in-line with the authors’ previous works, we concentrate on the so called 1

ASDR or equivalently known as “Reentry Disposal (RD)” missions. Up until now different concepts have been proposed to conduct such missions like using LASER beams [5,6] Space Tethers [7] and ASR as described in the current work. Design considerations and mission description required to deorbit launcher stages have been discussed thoroughly in [8]. Such missions are not seen to be economically feasible. Preliminary studies show cost to benefit to complete the mission are well beyond normal engineering works. With Reference [9] and a noncooperative TSO, phases of a typical mission include: 1- Pre-capture 2- Contact 3- Post-capture and finally 4- Compound stabilization, making ASR ready for its next rendezvous. Obviously, overall system must be observable and controllable throughout the mission. This, in turn, necessitates having proper knowledge (estimate) of all relevant IPs. IPs include: Mass, Center of Mass (CoM) as well as associated moments of inertia of all ASR elements. We further note that IPs could vary as ASR captures (or releases) a TSO during its mission. It is further emphasized that, it is not just enough to capture a TSO as we still need to guide the ASR to the de-orbiting position and to guide the captured TSO to its deorbiting position, we need to generate suitable commands that in-turn requires a good estimation of the inertia terms associated with the combination of ASR+TSO bodies or otherwise we just end-up creating a bigger debris in the space. It is evidently clear that for a precise rendezvous in space, one cannot possibly rely on the so called “Trial & Error” to avoid missions, like rendezvous or ASDR, be time consuming or costly. We need to develop techniques that are mathematically efficient and fast to have a powerful command and control for ASR to capture and deorbit the TSO. This work is an effort in that direction. In fact, the idea is to develop a general conjugate motion-model for an ASR which is both linear and modular that could significantly enhance the inertia estimation process while

2

ASR is capturing other objects. A fast Estimation, in-turn is expected to enable ASR to generate precise control commands throughout its mission. As it is seen, the current work falls in the area of In-Space “System Identification” (SI)”. So considering the nature of the space environment peculiarities, we next examine related SI works available in the literature. 1.1

Related Works

We start with Reference [10] that provides a comprehensive overview of all SI techniques. Nonetheless, we are only interested in the ones applicable to the space environment and those specifically related to the space-robots. However, this reference gives an excellent but general overview of all advances together with their shortcomings in the field. In the same line of work, [11] explains the basic concepts regarding “robots dynamics”. For the specific cases of “space robots” [12] and [13] developed Generalized Jacobian Matrix and Barycentric Vector to simplify the dynamic model for free-floating space-robots. Other works like [14] are excellent references in describing kinematic model of a space-robots with multiple manipulators. The problem of Dynamic/Control interaction for flexible space-robots is discussed in [15]. Ref. [16] presented a combined control strategy to avoid active control of base while maintaining the fuel consumption as low as possible. Other references like [17] and [18] discussed about the trajectory planning of the space-robot in pre-capture phase. The same problem studied by [19], where the trajectory of a flexible manipulator has been optimized to reach the minimum relative velocity at contact. A complete review on modeling and trajectory planning of space robots can be found in [20]. However, such references, although good in describing physical models, require knowledge of dynamics of the space-robot and grasped object. There are many different problems related to the system identification in space applications like attitude identification [21] of spacecraft’s, 3

identification of flexibility parameters of the spacecraft’s [22] but in this paper the focus is on IPs identification of space systems. There are also quite a few works related to the ground-based robots identification which we briefly examine to highlight their common shortcomings. Starting with [23] that offers a linearized form of equations of motion w.r.t the IPs of a typical ground robot. A thorough examination of [23] reveals that it uses a Newtonian approach that eventually falls short to accurately estimate IPs of the robot as the developed formulation, based on Newton-Euler formulation, needs full force-torque measurements. In the same line of work, Reference [24] use the linear form of [23] in conjunction with Singular Value Decomposition (SVD) technique and concludes that all parameters of a ground robot are not identifiable. However, it suffices to that fact that there is always a minimal set containing of independent inertial parameters which themselves are sufficient to identify/control robot’s behavior. It should be noted that, in space applications the base of the space-robot is usually floating which yields in different identifiability characteristics compared to that of a ground robots. Reference [25] followed by many other researchers have argued that space-robots exhibits nonlinear properties w.r.t their IPs. These nonlinearities appear especially where equations are being expressed in a Cartesian system. There are a few works that have tried to overcome this problem. In this regard, Reference [26] summarizes methods to identify the IPs of a spacecraft in three general categories of: 1- Vision-Based Approach (VBA), 2- Momentum-Based Approach (MBA) and finally 3- Force-Based Approach (FBA). In comparison, VBA relies on the fact that each point on the surface of an object should follow a trajectory which is constrained by the object’s dynamics. Therefore, having the trajectory of a given point, through using a laser camera, one could estimate the dynamic characteristics of the target object. There are quite a few

4

references that have used this approach to identify IPs of a rigid body travelling in Space [27,28]. It is noted that the three aforementioned techniques provide different results. For example, VBA advantage lies in the fact that the spacecraft need not to be in contact with the selected TSO. However, studies show that in this approach the mass and the absolute value of moments of inertia of the TSO are not observable [28]. In comparison to other techniques, MBA is an effective approach in cases where the changes in momentum of the space-robot body as well as its velocity are measurable. In such cases, we could estimate the IPs of the robot in a straightforward manner ([26,29,30] ). In fact, MBA has the advantage of not requiring accelerations measurement. It should be noted that accelerations measurements are relatively more susceptible to noise [26]. Nonetheless, the ability of the technique in the case of a freeflying-robot is still an open issue. The FBA approach, on the other hand, uses the relationship among actuation, inertia and the body responses to identify the IPs. Different case-studies suggest that FBA is less restrictive as opposed to that of VBA and MBA; when it comes to modularity concept pursued by the authors. Therefore, in this work we have used FBA in an effective manner to reach a linear-modular formulation. To describe the contribution of the current work, we need to thoroughly describe the details of FBA. This approach followed by many researchers to identify IPs of a fix object in wide variety of applications, see Ref. [31] and [32]. For example, in [32] authors adopt the approach to align the spacecraft thruster with that of space-debris’ center of mass. Identifying the IPs of varying configurations like robots is more complex as finding good identification formula is not straightforward. For example, Ref. [25] uses FBA to develop a linear form of equations of motion w.r.t the IPs of an unknown body captured by an ASR using just one arm and shows how IPs of the captured body could be identified. This reference has gone further to compare its

5

results with MBA, as a separate technique. Reference [33], has extended this work for a robot that has two arms and discusses how the two jointed bodies could be identified with a technique referred to as “Sequential Quadratic Programming (SQP)”. In this line of thought, [34] presents a method to simultaneously identify IPs of the ASR-base and its outmost manipulator while it considers an ASR with multi-arm manipulator. As a summary, we must emphasize that FBA usually requires more measurements than that in MBA. On the other hand, MBA uses a fixed number of governing equations and so it is used to estimate IPs of just one object (i.e. payload or base). In FBA the number of governing equations is proportional to the number of robot elements which provides a much better results and convergence speed. Although many works have used such technique, nonetheless, there are still considerable shortcomings with the MBA abilities to simultaneously estimate inertia parameters. A complete comparison between the identification approaches are presented in table I. All different references introduced so far, fundamentally fall into the category of identifying spacecraft IPs. However, there are other references like [35] that deal with typical robot control strategies therefore require SI techniques to reduce the associated mathematical model uncertainties. Such references focus to attain better qualities for “system states” and give little priority to the accuracy of identifiable parameters. Table I – Comparison of identification approaches in space applications Method VBA

FBA

Disadvantage  Can be applied to identify just one object  Cannot estimate mass  Estimates relative values of inertia matrix terms  Computational workload  Needs measurement of force and acceleration of all elements  Needs contact

Advantage  Estimates Shape and States of the object  Is contactless

 Can be applied to both rigid bodies as well as reconfigurable robots  Can be used to identify IPs of all elements of the robot.  Number of identification equations increased as the number of elements

6

increased MBA

   

 Needs less measurements  Simple equations

Cannot be applied to identify rigid bodies Limited number of identifiable elements Constant number of identification equations Needs contact

Current work, as a continuation of previous work of the authors [36], fall into the category of simultaneous estimation of space-robot IPs. Comparing to the previous researches on ground robots, the current approach is based on Lagrange formulation and so it employs all dynamic equations which are not dependent on contact force/moments. In addition, we aim to consider the criticality of the time needed to reach to viable estimations. That is for space operations all steps must be conducted in a timely fashion and so it is essential to identify the system to issue successive controlling commands. Next section describes the contribution of the current work. 1.2

Contribution of the Current work

As described earlier, it is necessary to have an accurate estimation of the space-robots dynamics to avoid unpredicted behaviors or poor performance of the control system. It is clear that as soon as an object (TSO) with unknown mass properties is captured, a modified control strategy needs to be devised. It is further emphasized that ASRs that are expected to conduct ADR mission must be small enough to keep such missions cost-effective and so it is essential to develop algorithms that respect this limitation. So far most researches assumed that the spacerobot is completely known and estimated the IPs of one unknown object (like space debris). On the other hand, this work formulates precise and fast enough estimations specifically tailored for missions similar to that of ADR, considering that the all IPs of the spacecraft are not necessarily known. Here, we also need to know or estimate mass of the robot-base and the captured object to have an effective model for the control system.

7

Based on existing literature, we have used basics of the Force-Based Approach (FBA) and modify it to reach an effective estimation algorithm for cases where IPs of more than one element could be unknown. The developed formulation is, in fact, linear w.r.t the Mass and Inertia Matrix and we have been able to express a general set of dynamic equations for spacerobots which are 1- linearized in form and 2- expandable with respect to its inertia parameters. With such a set, any standard identification techniques similar to Recursive Least Square (RLS) [37] is then applied to identify the mass and inertia matrix of a typical ASR. The Lagrange formulation has been employed to derive the linearized form of equation of motion and so there is no need to measure contact forces. In the process of formulating the algorithm, a new mathematical tool referred to as Matrix Set is introduced that help linearize the Lagrange equations of motion of a space-robot to a linear form as far as mass and inertia matrix is concerned. Although, the formulation has been studied for ADR applications, it is equivalently useful for other complicated missions like “In-space assembly”.

2

Space-Robot Model Description With Fig. 1, we observe an ASR that has a distinctive base and number of arm-manipulators,

as its elements. The mass properties of elements could vary while ASR attempts to capture a target object. We further assume that all elements are rigid and it is the outmost arm that attempts to capture TSO. It is further assumed that gravity gradient induced-moment, joints friction and other external disturbances have negligible effects. Each element has a local coordinate system attached to its neighboring joint; the ith element has a mass equal to defined around its center of mass.

8

and inertia matrix of ,

We then seek to identify IPs of all ASR elements (base & arms) as a multi-body system with kinematic constraints represented by (1): ⃗

⃗⃗⃗

⃗⃗⃗⃗

⃗

(1)

Where ⃗⃗⃗ and ⃗⃗⃗⃗ are, respectively, the linear and rotational velocities associated with the ith element of the spacecraft,

and

are the Jacobians and finally, ⃗ is the state vector of the

spacecraft.

Fig. 1 - Schematic of an Autonomous Space Robot (ASR) capturing a TSO

We now notice that Fig. 1, in fact, shows a system of rigid bodies (ASR+TSO) in motion (orbiting the Earth) and as soon as ASR attaches to an object, a new element adds to the original form of the ASR. Therefore, we need to develop a mathematical tool that allows addition and subtraction in the equations of motion as the ASR might attach or detach itself to an element. Here, we show that with the help of a specific “Matrix Set”, we can achieve this goal. For this, we rearrange the nonlinear equations derived from well-known Lagrangian dynamics (2) to get a linear form w.r.t the mass and inertia matrix of all elements involved. Next section describes the details.

9

3

ASR Flight - a Linearized Mathematical Model For a multi-body consisting of rigid elements, travelling in a non-gravitational environment,

the potential energy of a rigid system is negligible and the equation of motion is given by (2): ( ̇) ⃗

We seek a kinematic energy term {∑

(

⃗

(2)

⃗

in the form of (3): ̇ {∑

⃗ )}

)} ̇

(

(3)

Analyzing (2) and (3) term by term and with proper mathematical manipulations, we could think of

) and rotational (

to be of two distinguishable parts of translational (

) terms that

is: (4) Therefore, ̇ [

] ̇

̇ [

] ̇

(5)

̇ [

] ̇

̇ [

] ̇

(6)

In the following a new mathematical tool called Matrix Set and related definitions and lemmas are presented which facilitates demonstration of equations of motion of multi-body system. 3.1

Matrix Set, Definitions and Lemmas

Theoretically, equations of motion of a single particle differ from the ones of that for a rigid body due to the fact that a rigid-body is a system of particles. Therefore, we need to use matrices for a rigid-body while for a single particle a simple vector is sufficient. The idea behind Matrix Set is that one can perceive a space robot as a “system of system” of particles. Therefore, as soon 10

as robot grabs (or attaches to) a new object, a new set of particles are added to its original form. Such concept provides a new and effective representation of a multi-body dynamic system as space-robots are. To further continue the manipulations, we need some specific definitions and lemmas as the following: Definition 1 A Matrix set, is a set of ⟦ ⟧

Matrixes ( ) which is presented in the form of (7).

{ |

}

(7)

Definition 2 – The product of a Matrix set and a n-dimensional vector is a matrix in the form of: [⟦ ⟧

]

[

]

(8)

Definition 3 – Inner product of a Matrix set and n-dimensional vector ⟦ ⟧ Where

[

(9)

’s are the components of . Matrix with respect to a r-dimensional vector ⃗ (

Definition 4 - Derivative of a [

]

is a vector defined as:

(⃗ )]): ⟦ ⃗⟧

{[

(⃗ )

]|

}

(10)

vector then a rectangular matrix [ ] is defined as:

Definitions 5 – If , is a

[ ]

[

Definitions 6 – For a symmetric

]

(11)

matrix A, the vector ⃗ is defined as

[

]

One can easily show that: 11

(12)

[ ] Lemma 1- Considering the scalar parameter 𝒴

(13) (⃗⃗ ) ⃗ ,

⃗

with ⃗ being a n-dimensional

vector and A being a symmetric matrix, then the derivative of 𝒴 with respect to ⃗ is: (

( ) )

[⟦ ⃗ ⟧

Lemma 2- If A is a Matrix set with all [⟦ ⟧ Lemma 3- If A is a

]

(14)

being symmetric matrices then

] ⃗

[⟦ ⟧

⃗]

(15)

matrix, and ⃗ is an n dimensional vector, both dependent on time

(t), then (

)

(16)

The proofs for aforementioned lemmas are presented in Appendix A. With these, we can proceed to develop the generalized formulation for ASR estimation process. 3.2

Generalized Formulation for ASR IPs Estimation

An ASR could definitely have many different parameters. The specific ones we are interested are related to its dynamics and are as the following: 

The mass vector ⃗⃗⃗ is a vector representing ASR elements mass ⃗⃗



[

]

The inertia vector is a vector including all ASR elements orthogonal inertia matrices [⃗⃗



⃗⃗⃗

⃗⃗⃗ ]

(18)

is a matrix set related to the translational velocity energy term ⟦ ⟧



(17)

{

|

}

(19)

Is a matrix set related to the translational velocity energy term

12

⟦ 

⟧

{[

̇] |

⃗

̇]

[

̇ ]]

[

]

Is a matrix set with elements of ⟦ ⟧



(21)

is a matrix with blocks of [



(20)

is a matrix related to the rotational velocity energy term

[ 

}

(22)

’s

{ |

}

(23)

is a matrix related to the rotational velocity energy term

( ̇)

[

̇]

[

̇]

[

Using Lemma-3, one could compute the time derivative of ⟦ ̇⟧

{⟦

⃗

̇|

⟧

̇ ]]

[

(24)

to be:

}

(25)

Substituting (5) and (6) in (2), we end-up with four different terms and with proper manipulations (See Appendix B and Table II) the ASR equations of motion which is suitable for the Estimation of IP’s is obtained as: ([⟦ ⟧

̈]

[⟦ ̇ ⟧

̇]

[⟦

⟧

̇ ]) ⃗⃗

( ( ̇ ))]

( ̇) ̇

̇ ] ⃗⃗ ̇

[⟦ ⟧

[ [⟦ ⟧

̇]

(26)

Equation 26 has two important properties firstly, it is linear w.r.t the mass and inertia matrix of ASR elements (⃗⃗⃗ and ). Secondly, it shows no limitation on number of elements. That is, the form of the equation allows elements to be added or subtracted from ASR. The (26) and associated properties are valid for to the systems with linear constraints described by (1)

13

Realistic scenarios reveal that some elements’ IPs are already known based on available computer 3D models, such data play a useful role during on-line estimation process. Besides, such data provide the opportunity to use well known Estimation methods like RLS which have known stability and convergence properties. Examining (26) in more detail reveal that dynamic constraints appear themselves through Jacobians terms (

and

). Such terms could be used to model different constraints like

revolute joints, prismatic joints and even complex structures like multiple manipulator spacecraft. The developed equation, in fact, remains the same as configuration of ASR changes. 3.3

Special Case of Generalized Formulation - Rigid Body

It is also interesting to see how (26) will look like for a single rigid object (i.e. i=1). For a planar rigid body with mass (m) and inertia term (Iz) actuated with force ⃗ and torque , the well-known governing equations are: ⃗̇

̇⃗

(27)

̇ ̇

(28)

Equations (27) and (28) can be re-written as: ̇ [⃗ ]

⃗ [ ] ̇

[ ̇

]

[

Comparing (26) and (29) shows that expression

] ̇

[⟦ ⟧

[ ] ⃗̇]

(29)

and velocity vector ([⃗

]

) both

appear as coefficients of ̇ , assuming dimensions remain the same, we could envision a new parameter as “Apparent Linear Velocity Matrix [ seen that both [ ] and ([⃗ ̇ [⟦ ⟧

⃗̈]

[⟦ ̇ ⟧

⃗̇]

[⟦

⟧

] ⃗ ̇ ])

]

[⟦ ⟧

⃗ ̇ ]”.

In a similar statement, it can be

) appear as the coefficients of mass. So one can introduce ([

]

to be “Apparent Linear Acceleration” matrix. In the same manner

14

apparent angular velocity and apparent angular acceleration can be defined as ̇ ( (⃗ ))]

and [

]

[ ̇]

⃗̇]

[[⟦ ⟧

(⃗ ̇ ).

In spite of linearity w.r.t mass and inertia terms, Jacobians carry parameters related to the elements Center of Mass (CoM) that cause nonlinearities. More details can be found in an analytical example provided by authors in [36]. It is noted that the CoM positions could also affects measurements of accelerations and actuations involved in different phases of a mission. Such nonlinearities cause difficulties to simultaneously identify CoM as well as mass and inertias. To overcome this matter one can separate the problem into two distinct estimation problems and then solve them sequentially [38]. One uses the last estimated mass and inertia matrix to calculate the regressor matrix and then estimates CoM while the other estimates the mass and inertia matrix using the last estimated CoM. Table II – Linearized form of different terms of dynamic equation term

4

[⟦

State driven term of rotational energy

⃗

(

Linearized form

State driven term of translational energy

⃗

(

name

⃗̇ ⃗̇

)

Time driven term of translational energy

)

Time driven term of rotational energy

[⟦ ⟧

⃗ ̇ ] ⃗⃗⃗̇

⟧

⃗ ̇ ] ⃗⃗⃗

[⟦ ⟧

⃗̇]

[⟦ ⟧

⃗ ̈ ] ⃗⃗⃗

( (⃗ ̇ ))

[⟦ ̇ ⟧

⃗ ̇ ] ⃗⃗⃗

(⃗ ̇ ) ̇

Steps in ASR Estimation Process In realistic scenarios, there are quite a few uncertainties in the process of spacecraft’s IPs

estimation. For example: 1- fuel consumption to conduct a maneuver and 2- Complex shapes of spacecraft’s arms which makes them hard to model in most computer programs and finally, 3Unknown IPs associated with space-debris. Although precise computer models of the spacecraft elements are helpful, they still carry errors as much as 5% as far as IPs are concerned. We would 15

like to emphasize that accuracy of the IPs which assumed to be known is very essential in the estimation process. Case studies conducted by the author’s shows that 5% inaccuracy in IPs assumed to be known can lead up to 11.2% error in estimation of unknown IPs. To reduce the problems associated with different uncertainties, we have developed an efficient algorithm depicted in Fig. 2: in which the fix IPs of spacecraft elements is improved during pre-capture phase and then by using the results together with the initial estimations of TSO parameters (from VBA), the variable IPs (like outmost arm and base) are estimated in postcapture phase. In this figure the solid lines show sequences while the dashed lines represent the data streams. At this stage, we embark on the capability of (26) for simultaneous estimation of IPs of the desired spacecraft to: 1- improve the accuracy of fix IPs of spacecraft elements in pre-capture phase and 2- estimating the unknown IPs of debris and base in post-capture phase. Partitioning (26) as:

[

]{

}

(30)

With ⃗ representing known IP’s and ⃗ representing unknown ones, (30) is rearranged as: [

]

[

]

(31)

Equation (31) is linear w.r.t the unknown IPs and could then be used to estimate the mass properties of base and debris in post-capture phase. The numerical detail of the process is further discussed in the following case-studies.

16

Start Pre-Capture

Approach to a new debris Update IPs of the S/C Estimate States and IPs of the debris using VBA

Can be captured?

No

Yes

Capture

Capture Post-Capture

Update IPs of the base and outmost arm

Control and Stabilization End

Fig. 2 – Proposed process for estimate on of Spacecraft IPs

5

Case Studies

5.1

Model Specifications

In this section, we study an ADR scenario during which we are required to estimate spacecraft’s fix IPs in pre-capture as well as base and debris IPs in the post-capture phase. A planar model of the spacecraft (Fig. 3) includes base and four hinged arms, is used. This model is complex enough to demonstrate the value of the work. The general behavior of changes in the IPs during the assumed scenario is shown in Fig. 4 and Table III for two different payloads (Table IV). In this scenario, as soon as a rigid TSO is captured, an abrupt change appears in mass properties of the outmost arm. An in-house computer code developed in C# is used to simulate the steps of the estimation processes which used a 4th order Runge-Kutta algorithm. All simulations have also been examined with the help of models developed within MSc ADAMS .

17

Fig. 3 – The spacecraft model and parameters

Base Mass (Kg) Base Izz (Kg.m2) b

Mf If

a

Mf I0

Arm Mass (Kg) Arm Izz (Kg.m2)

a

M0

Capture

a

Pre-Capture

b

a

Arm Izz (Kg.m2)

If

Base Izz (Kg.m2)

b

M0 b

Arm Mass (Kg)

Base Mass (Kg)

I0

Post-Capture

Fig. 4 – The general behavior of time variations of base and outmost arm

Table III – Parameters of the spacecraft Base Parameters

Debris Size Medium Heavy

2552 2552

7087 7087

2452 1925

Units: Masses (Kg), Inertia terms (

6900 6100

Arms Parameters 0.85 0.85

20.5 20.5

22.45 22.45

3.75 3.75

2 2

Outmost Arm Parameters 22.45 22.45

3.75 3.75

), Lengths (m), Angles (Degrees)

Table IV – Parameters of the spacecraft

Size

Official name

Mediu m

IRS-P2 R/B

Sat. num.

Izz (Kg.m2)

Mass (Kg)

23324

431.25

912

18

934.45 3822.4

435 8700

2 2

Heavy

CZ-2C R/B

8696.2 5

31114

3800

Table V – Initial estimate of unknown parameters of Fig. 4 Param. Value

5.2

M (Kg) 2552

Base I (Kg.m2) 7087

M (Kg) 15.2

Arms I (Kg.m2) 2.7

Outmost Arm + Debris M (Kg) I (Kg.m2) 22.45 3.75

Effects of IPs update

In This section we study the detailed implementation of (26) to improve the accuracy of IPs of elements uncertainties. The ASR is initially assumed to be at rest. Exciting the first joint by a

Joint Rates (Deg/Sec)

constant torque of 0.5 N.m (Arm 1 in Fig. 3) gives the results presented in Fig. 5.

4 2 0 -2

Arm 1 Arm 2 Arm 3 Arm 4

-4 0

5

10

15

20

25

30

Time (Sec)

35

40

45

Fig. 5 – Joint rate variations (Case I)

For the estimation purpose, the initial covariance matrix of RLS algorithm has been set to with forgetting factor[37] set to =0.99. The initial estimates of unknown mass properties, presented in Table V, exhibit up to 32% error w.r.t the true values. The results of estimation of mass and inertia matrix of both base and arms are presented in Fig. 6 and for different loadings in Fig. 7. The simulation has been continued for 50 seconds to clearly show the behavior and rate of convergence achieved. Figures clearly show that, even with a weak excitation, errors have been drastically decreased to below 0.2% in less than 30 seconds (3000 iterations). This is, in fact, due to the linear nature of the developed technique and (26). With Fig. 7, the performance of (26) in identifying elements IPs (including base and debris), in post-capture phase, has been examined for different payloads. It is important to see that the 19

payload size has no considerable effects on the convergence speeds and accuracies. Therefore, one could simply argue that proposed approach helps simultaneously estimate both debris and arms in the post-capture phase. That is, there is no need to treat the problem in two different steps, as previous works suggest. Next, we show that having a good estimate of spacecraft’s arm IPs (in pre-capture phase) will significantly increases the convergence speed in post-capture phase. 2600

Base Mass (Kg)

2500

24

2400

22 Base Arm 1 Arm 2 Arm 3 Arm 4

2300

20

Arm Mass (Kg)

26

18

2200 7000

2

Base Izz (Kg.m2)

Arm Izz (Kg.m )

6 6950 6900 4 6850 Base Arm 1 Arm 2 Arm 3 Arm 4

6800 6750 6700

10

20

30

Time (Sec)

2

0

40

Base Mass (Kg)

Fig. 6 - Time history of estimation of spacecraft elements 2600 2400

No Load Medium Load Heavy Load

2200 2000

2

Base Izz (Kg.m )

7100 7000 6900 6800

No Load Medium Load Heavy Load

6700 6600 6500

10

20

Time (Sec) 30

20

40

Fig. 7 - Time history of Base IPs estimation for different loadings

5.3

IPs Update in Post-Capture phase

In the post-capture phase the spacecraft should stabilize itself to compensate the disturbances due to the momentum exchange between TSO and ASR. In cases where TSO and ASR are of comparable size, the spacecraft may burn fuel to stabilize, that is, it will lose weight and so its IPs will change. On the other hand, the outmost arm’s IPs will also change as soon as TSO is captured. Therefore, here we study how (26) perform for such cases. For this study, we further assume that all joints have equal angular rates of 2 Deg/Sec and there is no physical limitations on joint rates. For both heavy and medium size TSO, the first joint is actuated by a torque equal to 0.5 N.m, which yields in joint rate variations as presented in Fig. 8 and Fig. 9 for a 20 seconds time window. These Figures show that dynamic responses will be fast after t=9 seconds and t=8

Joint Rates (Deg/Sec)

seconds respectively for medium size and heavy size payloads.

40 20 0 -20

Arm 1 Arm 2 Arm 3 Arm 4

-40 0

5

10

Time (Sec)

15

20

Joint Rates (Deg/Sec)

Fig. 8 – Variation of joint angle rates for medium-size TSO 40 20 0 -20

Arm 1 Arm 2 Arm 3 Arm 4

-40 0

5

10

Time (Sec)

21

15

20

Fig. 9 – Variation of joint angle rates for Large-size TSO

We now apply Equations 26 and 37 to estimate the IPs of the base and outmost arm. The results of estimation process with the initial covariance matrix set to

and forgetting

factor of =0.99 have been presented in Fig. 10 and Fig. 11 for medium-size as well as large-size TSO, respectively. As it was expected, the estimation process is slower for a large TSO, nonetheless, for both cases parameters are estimated in less than one second (100 iterations). Fig. 12 demonstrates the effect of initial covariance matrix on the convergence rate for the case of medium-size TSO. Fig. 12 suggests that increasing the elements of initial covariance matrix, increases the overall estimation convergence rate. However, the convergence behavior in time is smoother while this parameter is kept low. High angular rates of the system elements after t=9

936

3830

935

3820

934

Heavy Debris Medium Debris

3810

3800 1940

0

0.5

1

1.5

933

2

932 2460

Heavy Debris Medium Debris

1935

2455

1930

2450

1925

2445

1920

0

0.5

1

Time (Sec)

1.5

2

2440

Base Mass for M.D. (Kg)

3840

Base Mass for M.D. (Kg)

Base Mass for H.D. (Kg)

Base Mass for H.D. (Kg)

(Fig. 8) results in fast convergence of the estimation process at this time.

2

8800 435.4 435.2

8750

435 8700

434.8

Heavy Debris (H.D.) Medium Debris (M.D.) 8650 6120

434.6 6920 6910

6100 6900 6080

Heavy Debris (H.D.) Medium Debris (M.D.) 6060 0

2 Base Izz for M.D. (Kg.m ) Debris Izz for M.D. (Kg.m )

2

Base Izz for H.D. (Kg.m2) Debris Izz for H.D. (Kg.m )

Fig. 10 - Time history of estimation of base and outmost arm mass

2

Time (Sec)

22

4

6890 6880

Fig. 11 - Time history of estimation of base and outmost arm I zz Estimated Base Mass (Kg)

2600 2400 2200 2000

P0 = 102 I P0 = 104 I 6 P0 = 10 I 9 P0 = 10 I

1800 1600

Estimated Debris Mass (Kg)

1400 1200 1000 800 600 P0 = 102 I 4 P0 = 10 I 6 P0 = 10 I 9 P0 = 10 I

400 200 0

0

5

10

Time (s)

15

20

Fig. 12 – Estimation of outmost arm IPs for different initial covariance matrices

5.4

Robustness of Proposed Algorithm

For critical and time-sensitive cases, as space applications are, it is essential to study the robustness of the propose algorithm. We analyze the behavior through conducting a sensitivity analysis w.r.t the measurement noises. Obviously, comparing to the MBA the proposed estimation process is more susceptible to noise as it requires more measurements. Here, we assume each sensor measurement (e.g. Accelerometers, resolvers and tachometers) represent a random number with Gaussian distribution with zero mean. Time histories of estimation for different resolver measurement noise while ASR is capturing a medium-size TSO case are presented in Fig. 13. Considering the high resolutions of current resolvers, it is seen that the resolver “measurement errors” have limited effects on the estimation accuracy and performance, especially after only 10 seconds.

23

A similar study for “tachometer accuracy” on the estimation performance is presented in Fig. 14. Again it is observed that tachometer’s measurement errors have negligible effects on the estimation accuracy. It is noted that, joints acceleration cannot be measured directly unless we have mounted accelerometers on each ASR elements. However, considering the high quality outputs of installed tachometers, one might differentiate tachometers outputs to compute associated joints accelerations. The effect of joint acceleration error on estimation accuracy is presented in Fig. 15. Considering the current technology of accelerometers, it is seen that proposed algorithm could effectively be used to estimate of ASR IPs together with TSO. It is seen that estimation process has converged to its real values before t = 7 sec, resulting from fast joint changes (Fig. 9). Clearly a better excitation can improve the estimation convergence and accuracy. Estimated Last Arm Mass (Kg)

940 930 920 910 900 S.D. = 0.01 (Deg) S.D. = 0.05 (Deg) S.D. = 0.1 (Deg)

890 880

5

10

Time (Sec)

15

Estimated Last Arm Mass (Kg)

Fig. 13 – Effect of Resolvers measurement noise on estimation process 940

920

900

880

S.D. = 0.001 (Deg/s) S.D. = 0.003 (Deg/s) S.D. = 0.005 (Deg/s)

5

10

Time (Sec)

15

Fig. 14 – Effect of Tachometers measurement noise on estimation process

24

Estimated Last Arm Mass (Kg)

1000 800 600 400 S.D. S.D. S.D. S.D.

200 0

-200

0

5

10

Time (Sec)

15

= 10 g = 40 g = 70 g = 100 g 20

Fig. 15 – Effect of acceleration measurement errors on estimating Outmost arm mass (Case III)

5.5

Effect of Unknown Center of Mass

So far, we have tacitly assumed that an accurate estimate of the Center of Masses (CoM) position for all spacecraft elements is readily available. Nonetheless, this is not an attractive assumption as 1- one cannot rely on Computer-Aided-Drawings to support such an assumption and 2- TSO mass, once captured, could have distinctive effects on the outmost arm’s CoM position. Ref. [32] discussed thoroughly about the importance of accurate estimation of CoM in operations like ASDR. We, now investigate how the outmost manipulator’s CoM position error could affect the resulting estimation accuracy. In the absence of any viable statistical model, we just repeat the estimation process for different assumed CoM positions for the outmost arm (Fig. 16). The contours in Fig. 16 have been separated with a dark-thick line representing values for no-error (true CoM positions). It can be seen that different assumed center of masses for the outmost arm can still yield in acceptable estimations for base-mass. Nonetheless, contours are varying significantly between the specified limits, with a relative error from (-)140% to (+) 60%. Such studies reveal that a good initial value for CoM, is in fact, critical in the overall estimation process. The same study has been repeated for the outmost arm (Fig. 17), with the dark-thick contour representing CoM zero-position-error. Again, one could clearly see the importance of CoM 25

initial value on the accuracy of the estimations. Nevertheless, it is interesting to observe that the “error behavior” lead one to think of it as a “design tool”. That is, such studies could be used to design new class of autonomous space robots that are better identifiable during space-missions and less sensitive to initial values needed for the estimation process.

100

Base Mass Estimation Error (%)

60 40 20 0 -20 -40 -60 -80 -100 -120 -140

50

0

-50

-100

-150 2

2.5

Base Mass Estimation Error (%)

A ssume

1.2

Assumed Y cg (m)

1

0.8

0.6

0.4 2

3

Assumed X cg (m)

d X (3m cg )

3.5

0.4

0.6

A

1

1.2

ed Y c ssum

0.8

g

(m)

100

50

0

-50

-100

-150

0.4

0.6

0.8

1

Assumed Y cg (m)

1.2

Base Mass Estimation Error (%)

1.5

100

50

0

-50

-100

-150

Fig. 16 Effects of CoM error on the base-mass estimation

26

2

3

Assumed X cg (m)

0 -10 -20 -30 -40 -50 -60 -70 -80 -90 -100

-20 -40 -60 -80 -100 -120 0.4 1.5

0.6

As

sum ed

Y

(m )

Outmost Arm Mass Estimation Error (%)

0.4

0.6

0.8

1

1.2

2.5

1

cg

Assumed Y cg (m)

2

0.8 3

1.2

A

3.5

d me ssu

X cg

0

-50

3

2.5

2

Assumed X cg (m)

1.5

0

-50

-100

-100

3.5

) (m

Outmost Arm Mass Estimation Error (%)

Outmost Arm Mass Estimation Error (%)

0

0.4

0.6

0.8

1

1.2

1.4

3.5

Assumed Y cg (m)

3

2.5

2

1.5

Assumed X cg (m)

Fig. 17 Effects of CoM error on the outmost-arm-mass estimation

6

Conclusion The idea of this work revolves around a fast, generalized formulation to identify a typical

spacecraft equipped with robotic arms. The developed formulation can effectively facilitate different engineering operations in the space. We have been able to show that current formulation is superior to previous and allows a fast simultaneous estimation of all spacecraft’s elements without incorporating optimization techniques. Therefore, presented formulation could easily be modified to include identifying spacecraft’s equipped with multiple manipulators. This work, due to its modular nature, is equivalently applicable to different phases of a space-mission; 27

especially the ones associated with command and control algorithms. The explicit regressor form of the devised algorithm can be derived analytically for any spacecraft configuration which can even make the estimation process faster for individually designed configuration scenarios. This regressor form increases the robustness and reduces the computational burden. Different casestudies conducted by the authors reveal that having more equations in the estimation process, like we do, works much better in time critical phases, like compound stabilization phase. Further works by the author reveal that the estimation process, in spite of being linear w.r.t the mass and inertia matrices, cannot simultaneously be linearized w.r.t the CoM parameters. Therefore, we might think of conducting an estimation process that consists of two stages of 1Estimation of mass and inertia matrix and 2- Estimation of CoM position and then recursively, iterate between the two stages to reach a meaningful solution. Further research is definitely needed to show if such idea could work. It is noted that authors desire to avoid the complexity of having to deal with a nonlinear optimization problem, as other researchers do. Case-studies, presented in this work, were carefully selected to show the ability and power of the work for real-time applications; where an ASR is in serious doubts to capture an unknown space-object which might jeopardize its stability. It should be noted that better computer models could provide better a priori estimate and that, in turn, could further increase the convergence speed of the estimation process. Comprehensive case-studies conducted by the authors reveal that there many cases where mass and inertia estimation is highly dependent on the CoM position. In such cases, the current work, could effectively be used as a “design tool” to decrease such “dependency on the CoM position”. In fact, authors believe that such an approach is very effective to design much better space-robots. That is, robots with better identifiability for space works. As the final point,

28

authors wish to emphasize that current work could effectively pave the way for further enhancements and studies, such as: 

Effect of excitation strategy on the convergence accuracies



Effect of the Earth’s gravity gradient in the identification process



Effects of manipulators’ joint friction on the accuracy of estimation



Effects of measurement noises on the overall robustness.



Multi-manipulator capturing while more than one outmost arms attempt to capture a single TSO.

Appendix A - Proof of Lemmas A1 - Proof of Lemma 1: The derivative of the scalar parameter 𝒴 with respect to the vector as 𝒴 𝒴 ⃗

[ Which

[

𝒴

(A-1) ]

[

]

]. Considering (8) and (10) ⃗

⃗

[

⃗

⃗]

⃗

So Considering the symmetry of A one can find from the definition that its derivative to symmetry also. So ⃗ [

⃗

⃗]

[ [⃗

⃗

⃗] ⃗

]

(A-2) [

29

]

is

Comparing (A-1) and (A-2), the proof of the lemma will be completed. A2 - Proof of Lemma 2: Using (8) one can write:

[

] ⃗

[

] ⃗

⃗ [

⃗ ⃗ [⃗

⃗ ⃗ [

]

⃗]

⃗ ⃗ ]

[⃗

]

so [

] ⃗

[

⃗

⃗]

⃗

Finally, considering (8) this yields to [

] ⃗

[

⃗]

A3 - Proof of lemma 3:

( ⃗)

∑

∑( ̇

)

∑(

̇)

∑

∑( ̇

)

∑(

̇)

[∑( ̇

)]

[∑(

̇ )]

[∑

]

⃗

⃗

B - Derivation of Linearized Forms of Equations of Motion B1- State driven term of translational energy Considering that

⃗

[

⃗

⃗̇ ∑

⃗̇] ⃗̇

(

)⃗

̇

and using the Lemma 1 one can obtain

[

⃗

⃗̇] ⃗̇

Combination of the above equation with (8) and (20) yields in 30

⃗ ̇ ] ⃗⃗⃗

[

⃗

B2- State driven term of rotational energy Considering (6) and using the Lemma 1 [

̇]

⃗

̇

[

̇]

⃗

̇

For each term we have

{[

⃗ [

] ̇]

⃗

]}

[ ̇

{[

̇]

̇

[

̇ ]}

(B-2)

Using (13) one can obtain

[ [

[

⃗̇]

⃗̇] ]

[

⃗̇]

⃗

[

[

[

[

⃗̇]

⃗̇] ]

[

]

]

Considering (21) and substituting in (B-2) yields: [

⃗

⃗⃗

⃗⃗ ] ⃗ ̇

⃗⃗ [

⃗⃗

[

⃗⃗⃗

⃗⃗⃗

⃗⃗⃗ ] ⃗ ̇

⃗⃗⃗

⃗⃗⃗

⃗⃗

⃗⃗⃗

Considering the definition of inertia vector (18)

⃗

[[ ({

Which

[

] ] ⃗̇

[

] }

) ⃗̇

] Substituting 31

{ |

}

⃗⃗⃗ ] ⃗ ̇

] ⃗̇

[

⃗

And considering Lemma-2 it can be easily shown that ] ⃗̇

[

⃗

] ⃗̇

[

⃗̇]

[

B3 – Time driven term of translational energy Differentiating the translational energy term (5) with respect to the ⃗ ̇ yields: {∑

⃗̇

Considering the definition of

)} ⃗ ̇

(

one can expand

as

) }⃗ ̇

(

{∑

⃗̇

⃗̇

Considering Lemma 3 this equation can be rewritten in the form of: (

⃗̇

)

) }⃗ ̇

( ̇

{∑

{∑

(

̇ )} ⃗ ̇

{∑

) }⃗ ̈

(

It can be shown that: (

⃗̇

)

[

⃗̇

⃗ ̇ ] ⃗⃗⃗̇

⃗̇

[ ̇ ⃗̇

̇ ⃗ ̇ ] ⃗⃗⃗

̇ ⃗̇

[

And considering (7) and (8) the abstract form can be rewritten as (

⃗̇

)

⃗ ̇ ] ⃗⃗⃗̇

[

[

⃗ ̈ ] ⃗⃗⃗

B4- Time driven term of rotational energy To evaluate

(

⃗̇

) we first write

⃗̇

⃗̇

as

{∑

)} ⃗ ̇

( 32

[ ̇

⃗ ̇ ] ⃗⃗⃗

⃗̈

⃗̈

⃗ ̈ ] ⃗⃗⃗

So (

⃗̇ ⃗

)

⃗̇

(

⃗̇

⃗̇)

Considering equation (13) and using definitions of [ ] and ⃗ this equation can be rearranged as (

⃗

)

(

⃗̇]

[

⃗̇]

[

⃗̇] )

[

So (

⃗

)

(

[

⃗ ̇ ])

(

⃗ ̇ ])

[

[

⃗̇] ̇

[

⃗̇] ̇

Defining the matrix (⃗ ̇ )

[

[

⃗̇]

[

⃗̇]

[

⃗ ̇ ]]

It can be seen that (

⃗

)

( (⃗ ̇ ))

(⃗ ̇ ) ̇

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