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Feedrate scheduling of a five-axis hybrid robot for milling considering drive constraints

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Abstract

Characterized by high rigidity and precision, large working space, and reconfigurability, hybrid kinematic machines are widely used in the five-axis machining of large parts in situ. The feedrate is limited by the velocity, acceleration, and jerk of actuated joints in high-speed machining due to the nonlinear motion introduced by the use of revolute joints and parallel kinematic module. To achieve a good balance between the machining accuracy and efficiency, an offline feedrate-scheduling algorithm considering the drive constraints of a five-axis hybrid machine is proposed. By adding a dimension of the curve parameter, the feedrate profile expressed by a cubic uniform B-spline is mapped into a two-dimensional curve with the redefined control points. Then, the feedrate-scheduling process is completed by iteratively modulating the control points of feedrate profile. The velocity, acceleration, and jerk of actuated joints are calculated by the kinematic analysis for a dual non-uniform rational basis spline (NURBS) toolpath. Based on this, the feedrate constraint equations are derived considering the geometry and drive constraints. The scheduled feedrate profile remains constant in most parameter intervals, while it changes smoothly in transition intervals without violating constraints. Simulations and experiments are carried out on the TriMule600/800 machining platform, and the results validate the correctness and effectiveness of the proposed algorithm.

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Data availability

The data sets generated or analyzed during this study are included in the submission of this manuscript.

Abbreviations

u :

Curve parameter of the parametric toolpath

k :

The order of the parametric toolpath and the feedrate profile

P(u):

3D Spline curve of the cutter tip position (mm)

P ′(u), P ″(u):

the first and second derivatives of P(u) with respect to u

O(u):

3D spline curve of the cutter axis orientation

ψ(u):

2D spline curve of the attitude angle (rad)

q :

Actuated joint position vector (mm) and (rad)

\( \dot{\boldsymbol{q}},\ddot{\boldsymbol{q}},\overset{\dots }{\boldsymbol{q}} \) :

The first, second, and third derivatives of q(u) with respect to time t

x :

Cutter location vector in the operation space (mm) and (rad)

\( \dot{\boldsymbol{x}},{\boldsymbol{x}}^{\prime } \) :

the first derivative of x(u) with respect to time t and parameter u respectively

T s :

Interpolation period of the CNC system

F(u),\( {\left\{{\boldsymbol{F}}_i\right\}}_{i=0}^{M-1} \) :

The feedrate profile of cutter tip position and its 2D control points

F ′(u), F ″(u):

The first and second derivatives of F(u) with respect to parameter u

δ, V ce :

Chord error and the maximum feasible feedrate

A, J :

Tangential acceleration and jerk of the cutter tip position

ω, V w :

Angular velocity and the maximum feasible feedrate

ω A :

Angular acceleration of the cutter axis

J a :

The [5 × 5] Jacobian matrix of the machining equipment

\( \hat{\boldsymbol{V}},\hat{\boldsymbol{A}},\hat{\boldsymbol{J}} \) :

Velocity, acceleration, and jerk vectors of the actuated joints

M :

Number of control points of the feedrate profile

U f :

Node vector of the feedrate profile

\( {\left\{{c}_i\right\}}_{i=0}^{M-1} \) :

The added parameter dimension of control points of the feedrate profile

F 1(u),\( {\left\{{\boldsymbol{F}}_{1,i}\right\}}_{i=0}^{M-1} \) :

initial feedrate profile considering process constraint and its 2D control points

V p, lim :

Specified process feedrate

F f(u), \( {\left\{{\boldsymbol{F}}_{f,i}\right\}}_{i=0}^{M-1} \) :

Feedrate profile considering the NI constraints and its 2D control points

F s(u), \( {\left\{{\boldsymbol{F}}_{s,i}\right\}}_{i=0}^{M-1} \) :

Feedrate profile after the iterative adjustment based on feedrate-constant interval and its 2D control points

\( {\left\{{u}_i^{\ast}\right\}}_{i=1}^{N_{\mathrm{sp}}} \) :

Parameter set of the sampling points

N sp :

The number of the sampling points

σ(u):

The change rate of the cutter axis orientation O(u)

κ(u):

Curvature of cutter tip position curve P(u)

\( {\left\{{\prod}_i\right\}}_{i=1}^{N_{\mathrm{fc}}} \) :

Feedrate-constant intervals

N fc :

The number feedrate-constant intervals

η :

Proportional adjustment coefficient

\( {\left\{{M}_{\mathrm{fc},i}\right\}}_{i=1}^{N_{\mathrm{fc}}} \) :

The number of control points related to interval ∏i

\( {\left\{{F}_{\mathrm{fc},i}\right\}}_{i=1}^{N_{\mathrm{fc}}} \) :

Constant feedrate value of interval ∏i

\( {\left\{{M}_{f,i}\right\}}_{i=1}^{N_{\mathrm{fc}}} \) :

The number of control points adjusted in the acceleration stage

\( {\left\{{M}_{b,i}\right\}}_{i=1}^{N_{\mathrm{fc}}} \) :

The number of control points adjusted in the deceleration stage

γ :

The difference between the ordinates of two adjacent 2D control points during the linear adjustment

\( {\left\{{M}_{\mathrm{ct},i}\right\}}_{i=1}^{N_{\mathrm{fc}}} \) :

The index of the last control point associated with interval ∏i

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Funding

This work is partially supported by the National Key R&D program of China (Grant No. 2017YFB1301800), National Natural Science Foundation of China (Grants 91948301, 51721003, and 51675369), Tianjin Science and Technology Program (Grant No. 17JCZDJC40100), and EU H2020-RISE-ECSASDP (Grant 734272).

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Authors and Affiliations

  1. Key Laboratory of Modern Mechanisms and Equipment Design of The State Ministry of Education, Tianjin University, Tianjin, 300350, China

    Guangxi Li, Haitao Liu, Wei Yue & Juliang Xiao

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  1. Guangxi Li
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  2. Haitao Liu
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  3. Wei Yue
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  4. Juliang Xiao
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Contributions

HL was in charge of the whole trial; GL wrote the manuscript; WY assisted with the process of analysis; and JX provided the assistance of theory. All authors read and approved the final manuscript.

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Correspondence to Haitao Liu.

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Li, G., Liu, H., Yue, W. et al. Feedrate scheduling of a five-axis hybrid robot for milling considering drive constraints. Int J Adv Manuf Technol 112, 3117–3136 (2021). https://doi.org/10.1007/s00170-020-06559-1

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  • DOI: https://doi.org/10.1007/s00170-020-06559-1

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