Method for judging stability of variable impedance model of robot system by using variable augmented matrix

Abstract

The invention belongs to the technical field of robot control, and particularly relates to a stability judgment method of a variable impedance model of a robot system using a variable augmented matrix, so that the stability of a robot can be verified offline when the rigidity of the robot is changed. Impedance control is a universal force control scheme suitable for various mechanical arm tasks, one application is man-machine cooperation, a mechanical arm conducts force and position mixed control on an object according to an impedance control law, an operator applies control force to the object or a robot to achieve man-machine cooperation, and work such as assembling and carrying is completed. According to different tasks of the mechanical arm, variable rigidity parameters can be used, and flexibility and compliance are embodied. Not only can the dynamic relationship between the external force and the robot motion be controlled, but also the dynamic behaviors can be flexibly changed in a continuous manner during the task period. Impedance control may be implemented in joint coordinates, Cartesian coordinates, or other task-specific coordinates. Impedance targets may be achieved in different ways.

Description

Technical Field

[0001] The invention belongs to the technical field of robot control, and in particular relates to a stability judgment method of a variable impedance model of a robot system using a variable augmented matrix. Background Art

[0002] Collaborative robots are a type of robot designed specifically to work with humans. Compared with traditional industrial robots, collaborative robots are more flexible and safer, and can work directly with humans in the same workspace without taking special safety measures.

[0003] Impedance control is a robot control strategy that aims to make the robot's response to external forces or torques similar to the impedance response in an electrical circuit. This control strategy allows the robot to flexibly adapt to changes in the environment, and is particularly suitable for situations where collaborative robots interact with humans or unknown environments. One application of impedance control is human-robot collaboration: the robotic arm performs force-position hybrid control on the object according to the impedance control law, and the operator applies control force to the object or the robot to achieve human-robot collaboration and complete tasks such as assembly and handling. Compared with constant impedance control, variable impedance control provides the flexibility to continuously adjust dynamics according to task requirements and environmental conditions. This adaptability enhances the capabilities of robots, enabling them to undertake more complex tasks in unstructured environments.

[0004] Through the study and understanding of these background technologies, this application proposes a stability judgment method for a variable impedance model of a robot system using a variable augmented matrix to solve the stability problem in variable stiffness impedance control of a robot arm, and proves its feasibility. Summary of the invention

[0005] The present invention aims to provide a method for determining the stability of a variable impedance model of a robot system using a variable augmented matrix.

[0006] In order to solve the above technical problems, the specific technical solutions of the present invention are as follows:

[0007] In some embodiments of the present application, a method for determining the stability of a variable impedance model of a robot system using a variable augmented matrix is ​​provided, comprising the following steps:

[0008] Step 1) obtaining state parameter information of the robot arm through joint sensors;

[0009] Step 2) constructing a desired variable resistance impedance-trajectory model;

[0010] Step 3) In the impedance controller, the user defines impedance parameters and analyzes the stability of the variable impedance;

[0011] Step 4) In variable impedance control, a state-independent stability condition related to variable stiffness is established by introducing a variable augmented matrix in the estimation of the derivative of the Lyapunov function and constructing an energy function by performing a weighted sum of the velocity error and the position error;

[0012] Step 5) Based on step 4), set the damping parameters and mass parameters;

[0013] Step 6) The posture error is calculated based on the posture information of the robot arm obtained by the sensor and the expected posture information of the robot arm. The control torque required for each task is calculated based on the error and the hierarchical variable impedance control model, and input into the robot arm to realize the control of the robot arm.

[0014] In some embodiments of the present application, the desired variable resistance impedance-trajectory model constructed in step 2) is:

[0015] The dynamic equations of a robotic arm with n degrees of freedom are as follows:

[0016]

[0017] Where M(q)∈R n×n is the inertia matrix in the joint space of the robot, is the Coriolis force matrix of the robot, G(q)∈R n×1 is the gravity term of the manipulator, J(q)∈R m×n is the Jacobian matrix of the robot. m is the dimension of the end effector task space, τ∈R n×1 is the joint input torque, where τ e is the external force on the joint;

[0018] In the joint space, the external torque τ of the mass-damper-spring system is e and position error It is expressed as:

[0019]

[0020] Among them, H∈R n×n , D∈R n×n , K(t)∈R n×n are the desired inertia, damping and variable stiffness matrices. H, D, K(t) are positive definite and symmetric;

[0021] Let τ c is the control torque, as shown in formula (3):

[0022]

[0023] Define the new control input μ as:

[0024]

[0025] Substitute equations (3) and (4) into (1) and introduce the symbol The closed-loop dynamics in equation (2) is obtained.

[0026] In some embodiments of the present application, in step 3), the stability of the variable impedance is analyzed using the Lyapunov function:

[0027]

[0028] Will It is abbreviated as V1, and V1 is differentiated along the trajectory of equation (2) by τ e =0, H and D are both constant matrices;

[0029]

[0030] The symmetry of the stiffness matrix eliminates and A mixed term, but containing The positive term still exists.

[0031]

[0032] In some embodiments of the present application, in step 4), the energy function is constructed by performing a weighted sum of the velocity error and the position error, a state-independent stability condition related to the variable stiffness is established, and a time-varying augmented free weight matrix is ​​introduced to reduce the conservatism of the energy function, and the Lyapunov function is constructed as follows:

[0033]

[0034] Where β(t) and γ(t) are symmetric, semi-positive definite, and continuously differentiable matrices. V2 is differentiated along the trajectory of equation (2), where τ e =0, and H is a constant matrix;

[0035]

[0036] To eliminate the inclusion of and , β(t) is defined as

[0037] β(t)=K(t)+αD+γ(t)K(t)-α 2 H.#(10);

[0038] Let D-αH+minK(t) / α≥0#(11), so that β(t) is

[0039] is semi-positive definite, and β(t) and Substituting into formula (9) we can get

[0040]

[0041] In some embodiments of the present application, in the stability condition under dynamic decoupling, it is defined as Theorem 1, let H and D be constant, symmetric, positive definite matrices; let K(t) be a symmetric, positive definite and continuously differentiable variable stiffness matrix, if there exists α>0, γ(t)≥0 and Make then exists is semi-negatively definite. is semi-negative definite, then the system in equation (2) is globally uniformly stable when τe = 0.

[0042] In some embodiments of the present application, in the first condition of Theorem 1, the degree of freedom provided by the free weighting matrix helps to reduce the conservatism of the stability judgment method, and γ(t) is selected to ensure that Theorem 1 holds.

[0043] The stability inequality derived from Theorem 1 gives Equation (13), which can be simplified to Equation (13) and Equation (11):

[0044] D-αH+min{Dγ(0),minK(t) / α}≥0.#(14)

[0045] Express the second condition of Theorem 1 as an inequality and move the term containing the time-varying free weight matrix to the left side;

[0046]

[0047] Integrating it in the t domain gives

[0048]

[0049] let have

[0050] Compared with the prior art, the present invention has the advantage that variable impedance control can not only control the dynamic relationship between external forces and robot motion, but also flexibly change these dynamic behaviors in a continuous manner during the task. Impedance control can be implemented in joint coordinates, Cartesian coordinates or other task-specific coordinates. Impedance targets can be achieved in different ways. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Various other advantages and benefits will become apparent to those of ordinary skill in the art by reading the detailed description of the preferred embodiments below. The accompanying drawings are only for the purpose of illustrating the preferred embodiments and are not to be considered as limiting the present invention. Moreover, the same reference symbols are used throughout the accompanying drawings to represent the same components. In the accompanying drawings:

[0052] Figure 1 A schematic diagram of a variable impedance stability offline verification block provided by an embodiment of the present invention;

[0053] Figure 2 A schematic diagram of a variable impedance control experiment provided by an embodiment of the present invention;

[0054] Figure 3 A schematic diagram of the mechanical arm error and energy change with the adjustment of the stiffness curve provided by the embodiment of the present invention-a;

[0055] Figure 4 A schematic diagram of the mechanical arm error and the energy changing with the adjustment of the stiffness curve provided by the embodiment of the present invention-b;

[0056] Figure 5 A schematic diagram showing the changes in stiffness, error, and energy of the variable impedance controller of a manipulator under external disturbance provided by an embodiment of the present invention;

[0057] Figure 6 Schematic diagram of the changes in stiffness, error, and energy of the variable impedance controller of the robotic arm provided in an embodiment of the present invention under external disturbances-b. DETAILED DESCRIPTION

[0058] The specific implementation of the present invention is further described in detail below in conjunction with the accompanying drawings and examples. The following examples are used to illustrate the present invention, but are not intended to limit the scope of the present invention.

[0059] In order to better understand the purpose, structure and function of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings.

[0060] See attached Figure 1-6 As shown, according to some embodiments of the present application, the following steps are included:

[0061] Step 1) obtaining state parameter information of the robot arm through joint sensors;

[0062] Step 2) constructing a desired variable resistance impedance-trajectory model;

[0063] Step 3) In the impedance controller, the user defines impedance parameters and analyzes the stability of the variable impedance;

[0064] Step 4) In variable impedance control, a state-independent stability condition related to variable stiffness is established by introducing a variable augmented matrix in the estimation of the derivative of the Lyapunov function and constructing an energy function by performing a weighted sum of the velocity error and the position error;

[0065] Step 5) Based on step 4), set the damping parameters and mass parameters;

[0066] Step 6) The posture error is calculated based on the posture information of the robot arm obtained by the sensor and the expected posture information of the robot arm. The control torque required for each task is calculated based on the error and the hierarchical variable impedance control model, and input into the robot arm to realize the control of the robot arm.

[0067] Among them, variable impedance control can not only control the dynamic relationship between external forces and robot motion, but also flexibly change these dynamic behaviors in a continuous manner during the task. Impedance control can be implemented in joint coordinates, Cartesian coordinates, or other task-specific coordinates. Impedance targets can be achieved in different ways, for example, actively on a torque-controlled back-drivable manipulator, or passively by placing elastic elements between the actuator and the joint. Through constant gain, impedance control makes the closed-loop system passive, and thus passive (and stable) in interaction with a passive environment. However, if the impedance parameters are allowed to change arbitrarily, the passive characteristics will be lost. Therefore, an analysis or control method that can ensure the stable execution of variable impedance tasks is needed. This application starts with the required variable impedance and hopes to verify whether it can be executed without losing stability.

[0068] Among them, the method includes: a robot variable impedance system stability verification method that does not rely on state variables, which can verify the stability of the system after the stiffness changes offline according to the expected change stiffness curve, and introduce a variable augmented matrix into the judgment method, which reduces the conservatism of the stability judgment method and increases the freedom of stability judgment.

[0069] Among them, the desired variable resistance impedance-trajectory model is constructed as:

[0070] The dynamic equations of a robotic arm with n degrees of freedom are as follows:

[0071]

[0072] Where M(q)∈R n×n is the inertia matrix in the joint space of the robot, is the Coriolis force matrix of the robot, G(q)∈R n×1 is the gravity term of the manipulator, J(q)∈R m×n is the Jacobian matrix of the robot. m is the dimension of the end effector task space, which is usually 6. τ∈R n×1is the joint input torque, where τ e The external force on the joint.

[0073] The goal of the variable impedance controller is to maintain the external torque τ e and position error The dynamic relationship between them. In the joint space, the external torque τ of the mass damping spring system e and position error It is expressed as:

[0074]

[0075] Among them, H∈R n×n , D∈R n×n , K(t)∈R n×n are the desired inertia, damping and variable stiffness matrices. H, D, K(t) are positive definite and symmetric.

[0076] Let τ c is the control torque, as shown in (3)

[0077]

[0078] Define the new control input μ as (4):

[0079]

[0080] Substituting (3) and (4) into (1) and introducing the notation The closed-loop dynamics in (2) is obtained.

[0081] In the impedance controller, the user-defined impedance parameters, such as the inertia matrix H, the damping matrix D, and the variable stiffness matrix K(t), determine the robot's resistance to external torque τ. e behavior when . If the selected impedance parameters are constant, the system is asymptotically stable for any symmetric positive definite impedance parameter matrices H, D and K. In this work, the present application focuses on variable impedance control. Specifically, the present application will assume that H and D remain constant, while K(t) is considered as a time-varying function. In addition, in order to implement an impedance controller, the external force τ e It is obtained through the joint torque sensor.

[0082] To analyze the stability of the variable impedance in (2), the following candidate Lyapunov function is considered:

[0083]

[0084] This application will It is abbreviated as V1. Differentiate V1 along the trajectory of (2) by τ e =0, H and D are both constant matrices.

[0085]

[0086] The symmetry of the stiffness matrix eliminates and A mixed term, but containing The positive term still exists.

[0087]

[0088] In (7), for negative semidefinite is negative semidefinite. Therefore, origin stability can only be concluded if all eigenvalues ​​of the stiffness matrix are constant or decreasing. Assume Increasing the stiffness eigenvalue injects potential energy into the system, and it is obvious that this will lead to unstable behavior.

[0089] As shown in (7), the classical Lyapunov energy function cannot determine stability when dealing with varying stiffness. As observed in (7), an obvious approach to ensure the stability of the system is to design a new controller that can suppress the energy injected into the system due to the varying stiffness. However, this approach has several disadvantages, the most important of which is

[0090] The allowed range of the variable impedance curve is determined by the robot's state, which cannot be predicted in advance;

[0091] 2) There is no guarantee that the controller can implement the desired change stiffness.

[0092] Experimental evidence of variable stiffness control shows that reasonable variable stiffness parameters usually do not lead to instability tendencies under general circumstances. This prompted researchers to explore Lyapunov candidate functions that are less conservative than (5). In variable impedance control, the energy function is constructed by taking the weighted sum of the velocity error and the position error to establish a state-independent stability condition related to variable stiffness. At the same time, a time-varying augmented free weight matrix is ​​introduced to reduce the conservatism of the energy function. The Lyapunov function is constructed as follows:

[0093]

[0094] where β(t) and γ(t) are symmetric, semi-positive definite, and continuously differentiable matrices. Differentiate V2 along the trajectory of (2), where τ e =0, and H is a constant matrix

[0095]

[0096] To eliminate the inclusion of and , β(t) is defined as

[0097] β(t)=K(t)+αD+γ(t)K(t)-α 2 H.#(10)

[0098] make

[0099] D-αH+minK(t) / α≥0#(11)

[0100] So that β(t) is positive semidefinite for all t>0.

[0101] Let β(t) and Substituting into (9), we can get

[0102]

[0103] Theorem 1: (Stability conditions under dynamic decoupling) Let H and D be constant, symmetric, positive definite matrices. Let K(t) be a symmetric, positive definite and continuously differentiable variable stiffness matrix. Then, if there exists α>0, γ(t)≥0 and Make

[0104] exist

[0105] is semi-negatively definite.

[0106] is semi-negatively definite.

[0107] Then the system in formula (2) is globally uniformly stable when τe=0.

[0108] Note 1: In the first condition of Theorem 1, the degrees of freedom provided by the free weighting matrix helps to reduce the conservatism of the stability judgment method.

[0109] Choose an appropriate γ(t) to ensure that Theorem 1 holds

[0110]

[0111] Then, the stability inequality derived from Theorem 1 yields (13), which simplifies (13) and (11) to

[0112] D-αH+min{Dγ(0),minK(t) / α}≥0.#(14)

[0113] Express the second condition of Theorem 1 as an inequality and move the term containing the time-varying free weight matrix to the left side.

[0114]

[0115] Integrating it in the t domain gives

[0116]

[0117] let have

[0118]

[0119] Note 2: (17) involves a free weighting matrix, which relaxes the constraints of the existing stability judgment method. All simplified conditional methods (14) and (17) contain γ(0). Choosing a larger γ(0) will undoubtedly increase the freedom restriction of the stability judgment criterion.

[0120] In practice, in impedance control, the stiffness parameter has the most significant impact on task performance. Therefore, it is usually prudent to prioritize stiffness design. Subsequently, the damping parameter can be selected to ensure stable execution of the desired variable stiffness, usually by using a critical damping parameter. Inspection of the constraints in Theorem 1 shows that the least conservative constraint is determined by the parameter α, so the least conservative constraint can be achieved by choosing the maximum eigenvalue of D-αH+min{Dγ(0),minK(t) / α} to remain negative throughout the execution. Therefore, to obtain the least conservative constraint, α should be chosen to be

[0121]

[0122] in and δ denote the maximum and minimum eigenvalues, respectively. With this choice, the first condition of Theorem 1 is essentially satisfied for all t. Therefore, only the second condition of Theorem 1 remains to be verified. A simple verification process for the stable impedance curve is as follows:

[0123] Select ideal H, D and K(t) matrices and determine the value of α according to equation (18).

[0124] Verify that for all t>0,

[0125] If the validation fails, improve the stiffness parameters and revalidate.

[0126] Under the second condition of Theorem 1, the larger the initial value of γ(t), the wider the range of variable stiffness that can be selected. This application selects the initial value of the time-varying free weight matrix γ(t) that exceeds minK(t) / αD. Therefore, (13) is transformed into D-αH+minK(t) / α.

[0127] The Gazebo platform was used to conduct a simulation experiment on the Franka robot arm, with the focus on the fourth joint of the Franka robot arm.

[0128] The system selects a mass parameter of 10Kg, a damping parameter of 1Ns / m, and a stiffness curve of K(t)=12+10sin(t). Figure 3 The curves of variable stiffness, joint error and energy function V2 are shown. The error depicted in the figure is divergent and the V2 energy curve shows upward fluctuations, clearly indicating unstable behavior. The stability criterion proposed in this application cannot determine the stability of the variable impedance system. Simply modifying the impedance parameters can make the system change from an unstable state to a stable state. Adjust the damping parameter to 3Ns / m and the variable stiffness to 18+10sin(t)e -0.02t As shown in Figure 4, the evolution of the adjusted variable stiffness, joint error and energy function V2 is plotted. The joint error and energy curves in the figure converge to zero, indicating that the system is stable. The stability criterion provided in this application confirms the stability of the adjusted variable stiffness curve. When evaluated using the general stability standard, the adjusted impedance parameters do not indicate stability because the variable impedance curve violates the second condition in the standard. It turns out that the general stability method is more conservative, while the stability method newly proposed in this application is more flexible and provides more degrees of freedom.

[0129] Figure 5 Stiffness, joint errors, and energy data were extracted over 75 seconds for an experiment with the variable stiffness set to K(t) = 15 + 10sin(0.1πt). The system was subjected to four external force disturbances, as shown in the shaded areas. A standard impedance controller was used to ensure that the actual stiffness matched the expected setting and to guarantee system stability. This result is confirmed by the drop in energy after the external disturbance was removed. (b) Same experimental parameters as (a), but the system uses an energy tank-based controller. When the energy in the energy tank is insufficient, the system cannot maintain the variable stiffness and has to use a constant stiffness value.

[0130] In the description of the present application, it should be understood that the terms "center", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside", etc., indicating orientations or positional relationships, are based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be understood as a limitation on the present application.

[0131] The terms "first" and "second" are used for descriptive purposes only and should not be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of the features. In the description of this application, unless otherwise specified, "plurality" means two or more.

[0132] In the description of this application, it should be noted that, unless otherwise clearly specified and limited, the terms "installed", "connected", and "connected" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection, or it can be indirectly connected through an intermediate medium, or it can be the internal communication of two components. For ordinary technicians in this field, the specific meanings of the above terms in this application can be understood according to specific circumstances.

[0133] In this specification, each embodiment is described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The same or similar parts between the embodiments can be referred to each other. For the device disclosed in the embodiment, since it corresponds to the method disclosed in the embodiment, the description is relatively simple, and the relevant parts can be referred to the method part.

[0134] The above description of the disclosed embodiments enables those skilled in the art to implement or use the present invention. Various modifications to these embodiments will be apparent to those skilled in the art, and the general principles defined in this application may be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention will not be limited to the embodiments shown in this application, but will conform to the widest scope consistent with the principles and novel features disclosed in this application.

Claims

1. A stability judgment method for a variable impedance model of a robot system using a variable augmented matrix, characterized in that: The following steps are involved: Step 1) obtaining state parameter information of the robot arm through joint sensors; Step 2) constructing a desired variable resistance impedance-trajectory model; Step 3) In the impedance controller, the user defines impedance parameters and analyzes the stability of the variable impedance; Step 4) In variable impedance control, a state-independent stability condition related to variable stiffness is established by introducing a variable augmented matrix in the estimation of the derivative of the Lyapunov function and constructing an energy function by performing a weighted sum of the velocity error and the position error; Step 5) Based on step 4), set the damping parameters and mass parameters; Step 6) The posture error is calculated based on the posture information of the robot arm obtained by the sensor and the expected posture information of the robot arm. The control torque required for each task is calculated based on the error and the hierarchical variable impedance control model, and input into the robot arm to realize the control of the robot arm.

2. The stability judgment method of a robot system variable impedance model using a variable augmented matrix according to claim 1, characterized in that: The desired variable resistance-trajectory model constructed in step 2) is: The dynamic equations of a robotic arm with n degrees of freedom are as follows: Where M(q)∈R n×n is the inertia matrix in the joint space of the robot, is the Coriolis force matrix of the robot, G(q)∈R n×1 is the gravity term of the manipulator, J(q)∈R m×n is the Jacobian matrix of the robot. m is the dimension of the end effector task space, τ∈R n×1 is the joint input torque, where τ e is the external force on the joint; In the joint space, the external torque τ of the mass-damper-spring system is e and position error It is expressed as: Among them, H∈R n×n , D∈R n×n , K(t)∈R n×n are the desired inertia, damping and variable stiffness matrices. H, D, K(t) are positive definite and symmetric; Let τ c is the control torque, as shown in formula (3): Define the new control input μ as: Substitute equations (3) and (4) into (1) and introduce the symbol The closed-loop dynamics in equation (2) is obtained.

3. The stability judgment method of a robot system variable impedance model using a variable augmented matrix according to claim 1, characterized in that: In step 3), the stability of the variable impedance is analyzed using the Lyapunov function: Will It is abbreviated as V1, and V1 is differentiated along the trajectory of equation (2) by τ e =0, H and D are both constant matrices; The symmetry of the stiffness matrix eliminates and A mixed term, but containing The positive term still exists.

4. The stability judgment method of a robot system variable impedance model using a variable augmented matrix according to claim 1, characterized in that: In step 4), the energy function is constructed by taking the weighted sum of the velocity error and the position error, and a state-independent stability condition related to the variable stiffness is established. At the same time, a time-varying augmented free weight matrix is ​​introduced to reduce the conservatism of the energy function, and the Lyapunov function is constructed as follows: Where β(t) and γ(t) are symmetric, semi-positive definite, and continuously differentiable matrices. V2 is differentiated along the trajectory of equation (2), where τ e =0, and H is a constant matrix; To eliminate the inclusion of and , β(t) is defined as β(t)=K(t)+αD+γ(t)K(t)-α 2 H.#(10); Let D-αH+minK(t) / α≥0#(11), so that β(t) is semi-positive definite for all t>0, and replace β(t) with Substituting into formula (9) we can get 5. The stability judgment method of a robot system variable impedance model using a variable augmented matrix according to claim 4, characterized in that: In the stability condition under dynamic decoupling, it is defined as Theorem 1. Let H and D be constant, symmetric, positive definite matrices; let K(t) be a symmetric, positive definite and continuously differentiable variable stiffness matrix. If there exists α>0, γ(t)≥0 and Make then exists is semi-negative definite. is semi-negative definite, then the system in equation (2) is globally uniformly stable when τe = 0.

6. The stability judgment method of a robot system variable impedance model using a variable augmented matrix according to claim 5, characterized in that: In the first condition of Theorem 1, the degrees of freedom provided by the free weight matrix help to reduce the conservatism of the stability judgment method. γ(t) is selected to ensure that Theorem 1 holds. The stability inequality derived from Theorem 1 gives Equation (13), which can be simplified to Equation (13) and Equation (11): D-αH+min{Dγ(0),minK(t) / α}≥0.#(14) Express the second condition of Theorem 1 as an inequality and move the term containing the time-varying free weight matrix to the left side; Integrating it in the t domain gives let have

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