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Power PMAC Mathematical Features November 2013

Power PMAC Mathematical Features November 2013. Power PMAC Floating-Point Math. Power PMAC processor has hardware floating-point engine 5–10x (+) faster than Turbo PMAC software-library implementation Single-precision (IEEE 32-bit) supported

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Power PMAC Mathematical Features November 2013

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  1. Power PMAC Mathematical FeaturesNovember 2013

  2. Power PMAC Floating-Point Math • Power PMAC processor has hardware floating-point engine • 5–10x (+) faster than Turbo PMAC software-library implementation • Single-precision (IEEE 32-bit) supported • Double-precision (IEEE 64-bit) supported, default for user programs • Double-precision provides much greater resolution and range than PMAC/Turbo PMAC 48-bit format • 18 more bits in the mantissa – 256K x increase in resolution/range • Range of ±1.80 x 10±308 • Suitable for key automatic motor calculations • Suitable for axis calculations (trajectories, kinematics, lookahead) • Suitable for user-program calculations

  3. Power PMAC Numerical Representations • Receiving numerical values in text strings (e.g. P1=3) • Standard decimal format • No arbitrary limit on number of decimal places permitted • Scientific notation • Format {mantissa}E{exponent} (e.g. -1.76E10) • Mantissa does not need to be normalized (e.g. -17.6E9 is OK) • Hexadecimal (with $ prefix) • Non-negative integers only • Reporting numerical values in text strings • Reporting format independent of receiving (e.g. hex vs. decimal) • Automatic decision on number of decimal places reported (formatting to fixed number of decimal places must be done in host computer) • When not an exact floating-point representation of fraction, full length is reported (~16 digits)

  4. Floating-Point Math in Automatic Routines • Motor phase and servo routines use floating-point math • Were fixed-point math in PMAC/Turbo PMAC • Speed and precision of floating-point math now suitable for these calculations • Many problems with fixed-point math eliminated • Register units are standardized (e.g. counts, milliseconds) • No position rollover worries • No velocity saturation worries • Minimal quantization noise introduced in “aggressive” filtering • No conversion overhead from axis calculations • Axis calculations use 64-bit floating-point math • Effective range before resolution-loss problems increased 256K times compared to Turbo PMAC

  5. Floating-Point Math in User Script Programs • Machine control applications require the use of many different numerical formats • Inputs and outputs are fixed-point (Boolean, 16-bit, 32-bit, etc.) • Most motion calculations are floating-point • Power PMAC Script environment does automatic type matching of different numerical formats • Fixed-point and floating-point representations • 1-bit to 64-bit formats • All “inputs” to calculations converted to 64-bit floating-point format • All mathematical operations done in this format • If calculation has a “result”, it is converted (if necessary) to required format • User does not need to concern himself with format conversions • (Note that user C-language programs must follow C rules for type matching and format conversions)

  6. Special Floating-Point Representations • Special representations that are part of IEEE-754 standard • Power PMAC reports these “values” with the text shown here • inf, -inf (+/-infinity) • Obtained (for example) by division by 0 • Operations that generate +/-inf do not create errors • inf > any finite number, -inf < any finite number, in a comparison • nan (not-a-number) • Obtained (for example) by function domain error (e.g. sqrt(-1) ) • Obtained when requesting value of element in non-existent hardware • Operations that generate nan do not create errors • Boolean isnan function can be used to check • -0 (minus zero) • Obtained by underflow of negative values • Equivalent to 0 (“plus zero”) in any subsequent operations • +/-inf, nan generally useful as diagnostics; in application, best to check before operation that could produce

  7. User Global Variables • “Global” variables can be shared between programs • P-variables (65,536 total) • Double-precision (64-bit) floating-point variables • Accessible from all coordinate systems • Declared as “global” (outside of a program) for IDE substitutions • Accessible from C programs with pshm->P[i] • Q-variables (8192 per coordinate system, not overlapping) • Double-precision (64-bit) floating-point variables • Separate coordinate systems access different variables for same name • Declared as “csglobal” (outside of a program) for IDE substitutions • Accessible from C programs with pshm->Coord[x].Q[i] • M-variables (16,384 total) • Pointer variables; take on format of what is pointed to • Accessible from all coordinate systems • Declared as “ptr” (outside of a program) for IDE substitutions • Not directly accessible from C programs (use API functions)

  8. Using M-Variables in Power PMAC • Most registers can be accessed without M-variables, by element name • M-variables with declared names can be convenient • Software (memory) registers rarely accessed with M-variables • Most users satisfied with standard data structure names • Some users want to shorten names • e.g. ptr Mtr1FE->Motor[1].PosError • Some users want to customize names • e.g. ptr LoadAxisEndPos->Coord[1].TPExec.Pos[6] • Hardware (I/O) registers commonly accessed with M-variables • Declared M-variable name will be meaningful for application • e.g. ptr LaserOn->Acc68E[1].DataReg[4].6 • Some I/O registers have no element name, defined by address • e.g. ptr LightCurtainBroken->u.io:$D00004.10.1

  9. Defining M-Variables • Defined to data structure element (takes on format of element) M{constant}->{data structure element} • General definition in I/O space M{constant}->{format}.io:{adr_offset}[.{start}[.{width}]] • General definition in user shared memory space M{constant}->{format}.user:{adr_offset}[.{start}[.{width}]] • Self-defined (useful to create particular numerical format) M{constant}->*{format}[.{width}] • M-variable formats • s – signed integer that saturates • i – signed integer that rolls over • u – unsigned integer that rolls over • f – short (32-bit) floating-point (does not take {start} or {width} ) • d – long (64-bit) floating-point (does not take {start} or {width} ) • {start} = 0 to 31 (default is 0, not limited to 1 & multiples of 4) • {width} = 1 to 32 (default is 32, not limited to 1 & multiples of 4)

  10. Power PMAC I-Variables • “Legacy” feature intended for convenience of experienced Turbo PMAC users • Point to Power PMAC data-structure savable setup elements that are essentially equivalent to present Turbo PMAC I-vars: • For Motors 0 – 31: I0 – I3199 • For Coordinate Systems 0 – 15: I5000 – I6599 • (Note that I0 – I99 taken for Motor 0, I5000 – I5099 for C.S. 0) • For “Gate1” ICs 0 – 19: I7000 – I7999 • For “Gate2” ICs 0 – 3: I6800 – I6999 • Usable in programs and on-line commands • Setup elements for new features, motors, C.S.’s generally not given I-variable numbers • Can find out what element is used by an I-variable with on-line command I{constant}-> e.g.: I428-> I428->Motor[4].InPosBand

  11. Power PMAC User Local Variables • “Local” variables are used within a program • L-variables (8192 per C.S., PLC, & on-line command processor) • Double-precision (64-bit) floating-point variables • Not (directly) accessible outside of program where used • Declared as “local” (inside of a program) for IDE substitutions • R-variables are renamed L-variables for passing/returning values to/from general subroutines • Rnof calling routine equivalent to Ln of called routine • Permit true “stack” argument passing to subroutines • C-variables are renamed C.S. L-variables for passing/returning axis values to/from kinematic subroutines • Cn is equivalent to L(MAX_MOTORS+n) for C.S. • C.S. and PLC data structures provide full read access to local variables • Coord[m].Ldata.L[n] or PLC[m].Ldata.L[n] accesses Ln relative to L0 of present program • Coord[m].Ldata.Lindex or PLC[m].Ldata.Lindex accesses number of L0 of present program relative to root L0 • Coord[m].Ldata.R[n] or PLC[m].Ldata.R[n] accesses Rn relative to R0 of present program

  12. Local Variable Stack Example • This example uses default “stack offset” of 256 • Other offsets can be declared • IDE declares optimal offsets automatically • Rn of calling routine equivalent to Ln of called routine • Rn of calling routine equivalent to L(n + StackOffset) of calling routine • Pass/return arguments should start with R0 of calling, L0 of called • IDE manages these definitions and assignments automatically

  13. Local D-Variables • “Non-stack” set of local variables for each C.S., PLC, communications thread • Separate from L, R, & C “stack” local variables • 53 variables (D0 – D52) for each environment • Used for “read” argument passing from letter/number format • Especially useful for RS-274 “G-Code” CNC-style programs • Not intended for multiple levels of subroutines (no stack) • Used for storing queried axis values • From commands like pread, dread, vread, fread • D0 used for kinematic routine input/output flags • Can be accessed “externally” by structure name • e.g. Coord[x].Ldata.D[n], Plc[i].Ldata.D[n]

  14. Auto-Assigning User Names to Variables • Requires use of Project Manager in IDE • When you don’t care what variable is assigned, use declarations • global MyPvar1, MyPvar2, MyPvar3(16); • csglobal MyQvar1, MyQvar2, MyQvar3(256); • ptr MyMvar1->{definition}, MyMvar2->{definition}; • local MyLvar1, MyLvar2, MyLvar3(7); • Project manager automatically assigns variables to names • Can set starting variable numbers for auto-assign with PVARSTARTn, QVARSTARTn, MVARSTARTn directives • Directives in /usrflash/project/pp_proj.ini • Defaults are PVARSTART8192, QVARSTART1024, MVARSTART8192 • Variables with lower numbers available for manual assignment or “raw” use • Local variable auto-assigns always start with L0 • L0 to L(n-1) for n arguments into routine, Lnand up for internally declared

  15. Capabilities with Declared Variable Names • When writing programs, just use declared variable names, e.g.: MyPvar1 = MyMvar2 + 7; • Can access from IDE terminal, watch window, etc., e.g.: MyPvar1 &2MyQvar3 MyMvar2 = 1 • If non-local variable declared with a value, e.g.: global MyPvar4 = 3; • Value is automatically assigned to the variable on download, power-up, and reset • Very useful for setting power-up default values • Value can be expressed as constant or expression (e.g. 5 * 65536), but expression cannot use variables • Communications program gpascii -2 can use declared variable names

  16. Manual Assigning User Names to Variables • Requires use of Project Manager in IDE • When you do care what variable is assigned, use definitions • #define MyPvar1 P10 • #define MyQvar7 Q50 • #define MyMvar3 M200 • Simple text substitution on download • Should use variables numbered below P/Q/MVARSTART # • Communications program gpascii -2 can use defined variable names • Can be most important for local (L) variables to coordinate variable passing properly (IDE does automatically for subroutine arguments) • #define MyLvar1 L0 • #define MyLvar2 L1 • #define MyRvar1 R0 • #define MyRvar2 R1

  17. Power PMAC Variable Arrays • For any numbered variable type (P, Q, M, I, L, R, D, C) • Specific variable can be specified by mathematical expression in parentheses • e.g. P(P1-P2+7), L(Index-1) • Technically, these are function calls, so use ( ), not [ ] • Compare to indexed data structures, which use [ ] for index • Computed expression value rounded down to next integer (if necessary) before used to select particular variable • Usable in motion and PLC programs, & many on-line commands • Can use arrays with variable declarations in IDE • global MyParray(32) declares 32-element P-variable array • If IDE project manager assigns P620 – P651 to this array, MyParray(Index) becomes P(620+Index) • No protection against index exceeding declared array length • Note distinction between arrays, vectors, and (2D) matrices

  18. Power PMAC Script Math: Scalar Functions • Trig functions using radians: sin, cos, tan, sincos • Inverse trig functions using radians: asin, acos, atan, atan2 • Trig functions using degrees: sind, cosd, tand, sincosd • Inverse trig functions using degrees: asind, acosd, atand, atan2d • Hyperbolic trig functions: sinh, cosh, tanh • Inverse hyperbolic trig functions: asinh, acosh, atanh • Log/exponent functions: log (or ln), log2, log10, exp, exp2, pow • Root functions: sqrt, cbrt, qrrt, qnrt • Rounding/truncation functions: int, rint, floor, ceil • Random number generation : rnd (32-bit), randx (64-bit), seed • Miscellaneous functions: abs, sgn, rem, madd (mult & add), isnan

  19. Power PMAC Script Vector & Matrix Functions • Treat sets of consecutively numbered variables as 1D vectors or 2D matrices for mathematical operations • Global P-variables if Ldata.Control = 0 (power-on default) • Local L-variables if Ldata.Control = 1 • Global user shared memory Sys.Ddata[i] variables if Ldata.Control = 2 (new in V1.5) • Particular variables used in vector or matrix specified by function arguments: • Starting variable number • Length for vectors • Number of rows and columns for matrices • Functions permit more compact user code • Functions execute much faster than multiple user code scalar operations

  20. Power PMAC Script Math: Vector Functions • Operate on continuously numbered set of variables • Same set of variables can be treated as 2D matrix for other purposes • Functions that produce new vectors (returned value is “OK” flag – must “put” somewhere even if don’t use) • vadd: Adds 2 vectors together to produce 3rd vector • vcopy: Copies contents of vector into 2nd vector • vscale: Multiplies each vector element by common scale factor, result in 2nd vector • Functions that produce scalar (as returned value) only • sum: Adds a number of evenly spaced elements together (as for trace of matrix) • sumprod: Multiplies pairs of elements of 2 vectors together, adding products into returned value (as for dot product)

  21. Power PMAC Script Math: Matrix Functions • Operate on consecutively numbered set of variables • Elements in same row are consecutively numbered • Functions that produce new matrices (returned value is “OK” flag or determinant of result matrix – must “put” somewhere even if don’t use) • mmul: Multiplies 2 matrices together to create 3rd matrix • mmadd: Multiplies 2 matrices together, adds product to 3rd matrix • minv: Inverts a square matrix to create a 2nd matrix • mtrans: Transposes a matrix to create a 2nd matrix • msolve: Solves simultaneous set(s) of equations represented by square (coefficient) matrix and 2nd (constant) vector/matrix • Functions that produce scalar (as returned value) only • mdet: Calculates determinant of square matrix • mminor: Calculates specified minor determinant of square matrix

  22. Declared Vector and Matrix Names • In IDE, can declare array variables for vectors and matrices, e.g.: global MyVector1(3), MyVector2(3); // P-variables, declared outside program global MyMatrix1(9), MyMatrix2(9); local MyVector3(3), MyVector4(6); // L-variables, declared inside program Local MyMatrix3(9), MyMatrix4(36); • Function calls can then use declared names, e.g.: Ldata.Control = 0; // Use global P-variables MyResult = sum(&MyVector1(0), 3, 1); MyTest = vcopy(&MyVector1(0), &MyVector2(0); MyResult = mdet(&MyMatrix1(0), 3); MyTest = minv(&MyMatrix1(0), &MyMatrix2(0), 3); Ldata.Control = 1; // Use local L-variables MyResult = sumprod(&MyVector3(0), &MyVector4(3), 3, 3, 1); MyTest = vscale(&MyVector3(0), 2.5, &MyVector4(3), 3); MyResult = mminor(&MyMatrix3(0), 3, 1, 2); MyTest = msolve(&MyVector4(0), MyMatrix4(0), 6, 6); • Note that do not need to start with element 0 of vector/matrix

  23. Operators and Conditional Comparators • Arithmetic operators • + (add), - (subtract), * (multiply), / (divide), % (modulo) • Normal rules of algebraic precedence apply • Bit-by-bit logical operators • & (bit-by-bit AND), | (bit-by-bit OR), ^ (bit-by-bit exclusive OR) • ~ (bit-by-bit invert; unary operator – technically a function) • << (shift left), >> (shift right) • Conditional comparators • == (equals), != (not equals), ~ (approx. eq.), !~ (not approx. eq.) • > (greater than), < (less than) • >= (greater than or equal to), <= (less than or equal to) • Condition combinatorial operators • && (AND), || (OR) • Condition negation operator ! (NOT)

  24. Standard Assignment Operators • General syntax: {variable}{operator}{expression} • Expression following operator evaluated during program flow • Resulting value immediately used to set value of variable preceding operator • If motion program is looking ahead due to blending, buffered dynamic lookahead, and/or tool radius compensation, assignments can appear out of sequence with executed motion • Use synchronous assignments to delay actual assignment until resulting executed motion • Simple assignment: = (expression value written into variable) • Assignments with arithmetic operation: +=, -=, *=, /=, %= • Assignments with logical operation: &=, |=, ^= • Assignments with shift operation: >>=, <<= • Increment/decrement assignments: ++, --

  25. Synchronous Variable Assignments • Needed primarily because motion programs must calculate ahead during sequence of blended moves • Standard assignments would occur “too soon” compared to moves • Synchronous assignments are delayed to be in sequence with moves • General syntax: {variable}{synchronous operator}{expression} • Most variable types can receive synchronous assignment • Not just M-variables as in PMAC/Turbo PMAC • Including pre-defined data structure elements • No local variables or self-referenced M-variables can receive • Expression following operator evaluated during program flow • Resulting value stored in buffered queue along with operation type • Each coordinate system has an independent buffered queue • Actual assignment using operator occurs at beginning of execution of next commanded move (at start of blend) • In PLC programs, use to assure that commanded move has started

  26. Synchronous Variable Assignments (cont.) • Simple assignment: == (expression value written into variable) • For any type of variable format • Assignments with arithmetic operation: +==, -==, *==, /==, %== • *==, /==, %== for floating-point variables/elements only • Assignments with logical operation: &==, |==, ^== • For integer variables/elements only • Increment/decrement assignments: ++=, --= • Useful in PLC programs to know that commanded move has started, e.g.: P1=0; P1==1; jog1=100000; // P1 set at start of jog move while (!(P1) || !(Motor[1].InPos)) { } // Move not started or ongoing M1=1; // Set output when move has finished and settled // Motor must be assigned to a C.S. to trigger sync assignment in PLC

  27. Synchronous Assignment Motion Program Example • One move lookahead (for simple blending): X10; // 1st move X20; // 2nd move Q1 = 30; // Standard assignment M1 = 1; // Standard assignment M2 == 1; // Sync assignment X(Q1); // 3rd move • Q1 set to 30 at X=10 (t1) • Needs to be done at lookahead time to calculate next move properly • M1 set to 1 at X=10 (t1) • Output would appear out of sequence • M2 set to 1 at X=20 (t2) • Output would appear in sequence

  28. Synchronous Assignment Buffer • Synchronous assignments must be buffered from time encountered in program until delayed execution • Default buffer space for 1M (1,048,576) synchronous assignments • Size can be changed in IDE Project Manager (pp_proj.ini file) • Coord[x].SyncOps specifies how many of these assignments reserved for this C.S. in buffer • Default value of 8192 for each C.S. (sufficient for almost all applications) • To change buffer organization, set new values for Coord[x].SyncOps, save, and reset (new buffer offsets calculated automatically on reset) • No need to define second-stage buffer with special lookahead • C.S. needs to buffer for length of lookahead and retrace at any given time (buffer space is reused) • Large numbers of buffered assignments only needed for applications with special lookahead (and especially lengthy retrace)

  29. Power PMAC Script String Functions • Operate on character (byte) arrays in user shared memory buffer • Specify string variable with starting index value i in Sys.Cdata[i] • Functions that manipulate string variables: • sprintf, strcat, strcpy, strncat, strncpy, strtolower, strtoupper • Functions that interrogate string variables: • strchr, strcmp, strcspn, strlen, strncmp, strpbrk, strrchr, strspn, strstr, strtod • All functions return a value that must be “put” somewhere • Methods to use/transmit strings: • Script program send, cmd, system commands • Referenced within text by %s designator • Starting character index for string specified after text • e.g. send1"Error: %s“,512 • C routines and applications • Access string with character (byte) pointer into user shared memory buffer • Note that C can use strings elsewhere as well (but no Script access to these)

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