Energy optimization problem of robotic cells formulated as an Integer Linear Programming problem.

General description

Although the energy optimization problem of robotic cells is inherently non-linear, it is possible to formulate it as an Integer Linear Programming problem if the non-linear convex functions are piece-wise linearized. These functions correspond to energy functions of robot movements which express the dependency of the energy consumption on the duration of the movement for a fixed trajectory. A solution for one robot can be perceived as the most energy efficient closed path through the specified stop coordinates, called locations. During the waiting in a stationary robot position a power saving mode of a robot can be applied. If the whole robotic cell is optimized, time synchronizations and the spatial compatibility have to be met, i.e. a workpiece is handed over from one robot to another at the right place at the right time. Finally, as robots are close to each other, the collisions have to be avoided.

Table of used symbols

symbol	description	reference	type
	set of all activities, i.e. $V \cup E$	Activity	set
	set of all static activities	StaticActivity	set
	set of possible coordinates, i.e. locations, for activity $v \in V$	Location	set
	set of all dynamic activities	DynamicActivity	set
$E_\mathcal{O}$	set of all optional dynamic activities	DynamicActivity	set
	set of movements of activity $e \in E$	Movement	set
$\mathcal{R}$	set of robots	Robot	set
	power saving modes of robot $r \in \mathcal{R}$	RobotPowerMode	set
	static activities of robot $r \in \mathcal{R}$	–	set
	predecessors of activity $a \in A$	Activity::mPredecessors	set
	successors of activity $a \in A$	Activity::mSuccessors	set
	indices of the piece-wise linear functions	–	set
$k_{e,b}^t, q_{e,b}^t$	coefficients of b-th linear function	–	constants
$E_{\mathcal{T\hspace{-1.0mm}L}}$	time lags for inter-robot time synchronization	TimeLag	set
$l_{a_1, a_2}$	length of a time lag $(a_1, a_2) \in E_{\mathcal{T\hspace{-1.0mm}L}}$	TimeLag::mLength	constant
$h_{a_1, a_2}$	height of a time lag $(a_1, a_2) \in E_{\mathcal{T\hspace{-1.0mm}L}}$	TimeLag::mHeight	constant
$Q_{l_i}$	set of locations compatible with location	–	set
	multiple of the robot cycle time	–	constant
	either movement $e \in T_e$ or location $l \in L_v$	ActivityMode	mode
	set of possible collisions	RoboticLine::mCollisions	set
$\overline{W}$	upper bound on energy consumption	–	constant
	static activity that closes the cycle of robot $r \in \mathcal{R}$	Activity::mLastInCycle	activity
$p_{v,l}^m$	input power of robot $r \in \mathcal{R}$ for activity $v \in V_r$ and location $l \in L_v$	LocationDependentPowerConsumption	constant
$\underline{d}_e^t$	minimal duration of the movement $t \in T_e$	Movement::mMinDuration	constant
$\overline{d}_e^t$	maximal duration of the movement $t \in T_e$	Movement::mMaxDuration	constant

Table of variables

variable	description	reference	type
	energy consumption of activity $a \in A$	VariableMappingLP::W	positive float
	start time of activity $a \in A$	VariableMappingLP::s	positive float
	duration of activity $a \in A$	VariableMappingLP::d	positive float
	true if location $l \in L_v$ of activity $v \in V$ is selected, otherwise false	VariableMappingILP::x	binary
	true if mode $m \in M_r$ is applied in activity $v \in V_r$ , otherwise false	VariableMappingILP::z	binary
	true if movement $t \in T_e$ of activity $e \in E$ is selected, otherwise false	VariableMappingILP::y	binary
$w_{e,v}$	true if activity $e \in E_\mathcal{O}$ is performed $\left(v \in S_e\right)$ , otherwise false	VariableMappingILP::w	binary
	decides whether $a_1 \rightarrow a_2$ or $a_2 \rightarrow a_1$ to resolve a collision	VariableMappingILP::c	binary

Integer Linear Programming model

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Description of constraints

constraint(s)	description	reference
(1)	Minimization of the total energy consumed by all the activities.	RoboticLineSolverILP::construct
(2)	Propagation of the energy consumption of static activities to the criterion with respect to the selected locations, power saving modes of robots, and durations.	ConstraintsGenerator::addEnergyFunctions1
(3)	Propagation of the energy consumption of dynamic activities to the criterion with respect to the selected movements and robot speeds.	ConstraintsGenerator::addEnergyFunctions2
(4), (5), (6)	Only one power saving mode and location is assigned to each static activity, and each mandatory dynamic activity has assigned just one movement. Constraints (6) can be omitted as flow preservation constraints (7), (8) and the timing enforce the implicit selection.	ConstraintsGenerator::addUniqueModeSelection
(7), (8)	Flow preservation constraints ensure that if the robot enters to a location it also has to leave the same location.	ConstraintsGenerator::addFlowConstraints
(9), (10)	Ensure a correct timing for fixed precedences of activities	ConstraintsGenerator::addFixedPrecedences
(11), (12), (13)	Ensure a correct timing of optional precedences that model alternative orders of operations.	ConstraintsGenerator::addPrecedenceSelectionConstraints, ConstraintsGenerator::addSelectablePrecedences
(14), (15)	Restrictions on durations of static and dynamic activities, respectively.	ConstraintsGenerator::addDurationConstraints
(16)	Time synchronization between robots is ensured by time lags.	ConstraintsGenerator::addTimeLags
(17)	Constraints enforcing the spatial compatibility for handover operations.	ConstraintsGenerator::addSpatialCompatibilityConstraints
(18), (19)	Constraints ensure that two related activities are time disjunctive for different multiples of the production cycle time, and therefore, the collision is resolved.	ConstraintsGenerator::addCollisions
(20)	Domain of variables, all the variables of the model are either positive floats or binary variables.	VariableMappingILP

Remarks: In the implementation, constraints (6) are omitted, and some of constraints are merged, therefore the number related to a constraint (in the constraint description) differs from the numbers presented here. This mathematical formulation was adapted to ease the understanding and reading.